Elom Ayih Domlan
email: elom-ayih.domlan@ensem.inpl-nancy.fr

José Ragot

Didier Maquin

Switching Systems Mode Estimation Using A Model-Based Diagnosis Method

Keywords: switching system, estimation, mode, discernability

Switching systems are a particular class of hybrid systems. They are represented by several operating regimes, called modes, each of them being active under certain particular conditions. The modes can be associated with normal or faulty operating conditions. Therefore, the determination of the active mode at any moment constitutes a fault diagnosis approach for such systems. This paper addresses the issue of the determination of the active mode at any moment, using only the system's input/output data. To perform this task, we propose an adaptation of the well-known parity space method to this class of system. Conditions that guarantee the uniqueness of the determined active mode are also given.

Introduction

The modeling of complex systems often leads to complex non-linear models. To get rid of the obtained model's complexity, a widely used modeling strategy is to represent the system's behavior using a set of models with simple structure, each model describing the behavior of the system in a particular operating zone. Within this modeling framework, hybrid models are very successful in representing such processes. Hybrid models [START_REF] Heemels | Equivalence of hybrid dynamical models[END_REF] characterize physical processes governed by continuous differential equations and discrete variables. The process is described by several operating regimes, called modes, and the transition from one mode to another is governed by a mechanism which depends of the system's variables (input, output, state) or external variables (human operator for instance). When the transition from one mode to another is abrupt, one obtains a particular, but significant, class of systems namely switching models. This class of models is widely used because tools for analysis and control of linear systems are well mastered and, moreover, many real processes can be represented by models belonging to this class. Switching system [START_REF] Juloski | Hybrid Systems: Computation and Control[END_REF] are described by several operating regimes that are often linked to linear models. The transition from one mode to another is governed by a mechanism which depends on the system's variables (input, output, state) or external variables (human operator for instance). When the nature of the switching mechanism is unknown, the determination of the mode describing the behavior of the system at any mo-ment, this mode being called active mode, is a crucial information. We assume here that the modes describe the behavior of the real system either in normal operating conditions or in faulty operating conditions. The aim of this paper is to present a method for the determination of the active mode at any moment using the measurement data (input and output) of the system. This task is equivalent to a fault diagnostic scheme as it leads to the detection of the modes associated with faulty operating conditions. The determination of the active mode is performed thanks to an extension of the well-known parity space method [START_REF] Chow | Analytical redundancy and the design of robust failure detection systems[END_REF] to switching system. The paper starts in section 2 with the problem statement. The determination of the active mode is tackled in section 3. Conditions guaranteeing the discernability of the various modes are formulated in section 5 and an illustrative example is proposed in section 6.

Problem statement

Let us consider the switching system represented by equation (1):

x (k + 1) = A µ k x (k) + Bu (k) y (k) = Cx (k) A µ k ∈ A = {A 1 , A 2 , . . . , A s }, µ k ∈ {1, 2, . . . , s}, s ∈ N * \{1} (1)
In equation ( 1), the variables u(•), y(•) and x(•) respectively stand for the input, the output and the state of the system. The regime switchings are introduced by means of the state matrix which takes its value in a finite set A which is a priori known. The results presented in this paper can be extended to the case where the matrices B and C also take different values. The variable µ (•) denotes the active mode at any moment. For example, if one has µ k = i, i ∈ {1, 2, . . . , s}, the system is said to be in the mode i at the instant k. We assume that the switchings are triggered by unknown external variables and then, the mode sequence is arbitrary and independent of the system's variable (input, output and state). Coming from (1), we wish to recover the active mode (or the value taken by µ (•) ) at any moment, using only the system's input/output data on a finite observation window. We introduce the following definitions:

Definition 1 (Path) A path µ is a finite sequence of modes: µ = ( µ 1 • µ 2 • . . .• µ h ).
The length of a path µ is denoted |µ| and Θ h denotes the set of all paths of length |µ|. µ [i,j] is the infix of the path µ between i and j: µ

[i,j] = (µ i • µ i+1 • . . . • µ j ).
Definition 2 (Observability matrix) The observability matrix O µ,h of a path µ ∈ Θ h is defined as:

O µ,h =        C CA µ1 . . . C A µ h-1 A µ h-2 • • • A µ1 h-1        (2)
Definition 3 (Active path) On a finite observation window [kh, k], the active path µ * is the one describing the true mode sequence on the observation window.

From definitions 1 and 3, the estimation of the active mode at any moment is equivalent to the determination of the path describing the true mode sequence on a finite observation window. For that, throughout the remainder of this paper, we will focus on the recovery of the active path on an observation window.

Determination of the active path

The determination of the active path task can be formulated as a recursive problem applied to a sliding window. On a time window [kh, k], equation ( 1) can be written in a compact way as:

Y k-h,k -T µ,h U k-h,k = O µ,h x(k -h) (3)
where

Y k-h,k = y (k) . . . y (k -h) T , U k-h,k = u (k) . . . u (k -h) T and
T µ,h is a Toeplitz matrix defined by:

T µ,h =        0 . . . 0 0 CB 0 0 . . . . . . . . . C A µ k-1 . . . A µ k-h+1 h-1 B . . . CB 0        (4)
The relation (3) links on the time window the system's input and output to the initial state x(kh) oon the observation window. We introduce the following proposition:

Proposition 1 The observability matrices O µ,h of the paths µ generated on the observation window [k -h, k] are all of full rank: rank (O µ,h ) = dim (x) = n, ∀h ≥ n.
The existence of an integer h, such that proposition 1 holds, has been analyzed in [START_REF] Babaali | Pathwise observability and controllability are decidable[END_REF] and is linked to pathwise observability that have been furthermore shown to be decidable.

Using proposition 1, a projection matrix1 Ω µ,h is defined in such a way that Ω µ,h O µ,h = 0, i.e. Ω µ,h is selected as a basis for the left null space of O µ,h . Next, residuals r µ,h (•), independent of the initial state x(kh), can be defined as:

r µ,h (k) = Ω µ,h (Y k-h,k -T µ,h U k-h,k ) (5)
The residuals r µ,h (•) depends only on the system's input and output and their calculation requires the preliminary determination of all matrices Ω µ,h .

Theorem 1 (Active path determination) The active path µ * describing the true mode sequence on a time window [kh, k] satisfies :

r µ * ,h (k) = Ω µ * ,h (Y k-h,k -T µ * ,h U k-h,k ) = 0 (6)
To recover the true mode sequence µ * from the system's measurements, one can proceed in the following way:

• first, all the possible paths of length h are built on the time window [kh, k]. This is equivalent to finding all the matrices O µ,h .

• knowing the matrices O µ,h , the projection matrices Ω µ,h are easily calculated.

• from the matrices O µ,h and Ω µ,h , one can form the residuals r µ,h (•) using the system's measurements.

• the active path is recovered from the system's measurements by testing the residuals r µ,h (•) and it corresponds to the one which residual is equal to zero.

Let us noticed that the same methodology can be applied for the determination of the active in the case of noisy measurement by allowing the test ( 6) to take into account the presence of noise on the system measurement [START_REF] Domlan | Systèmes à commutation : diagnostic de fonctionnement et identification de la loi de commutation, e-STA[END_REF]. Finally, once the active mode is identified, the state x(•) of the system can be estimated using a finite memory observer [START_REF] Ragot | Switching time estimation of piecewise linear system. Application to diagnosis[END_REF].

On the number of path

It is easy to see that the enumeration of all paths on a time window [kh, k] introduces a problem of combinative explosion related on the number of modes and the length of the observation window. Indeed, the number of residuals r µ,h (•), µ ∈ Θ h , to be calculated is equal to s h and quickly grows with the length h + 1 of the observation window and the number s of modes. Then, the use of all paths on a time window is awkward and computationally demanding. In practice, all paths µ ∈ Θ h do not have to be considered at every moment. When at a time k 0 , the active path on an observation window

[k 0 -h, k 0 ]
is identified, it is not necessary to test the s h residuals at the next instant k 0 + 1. Only the paths µ ∈ Θ h with infixes µ [k0+1-h,k0-1] identical to the infix µ * [k0+1-h,k0-1] of the path µ * recovered previously at k 0 are considered at the next instant k 0 + 1. Moreover, assuming that the minimum sojourn time in a mode is less than the length of the observation window, one can limit the number of generated paths by only considering paths that describe the mode sequence when the system remains in the same mode all over the duration of the observation window, i.e. µ = (i • i • . . . • i), i ∈ {1, 2, . . . , s}. Nevertheless, the reduction of the number of residuals comes at the expense of a delay in the estimation of the switching time from one mode to another. The recognition of the active path cannot take place as long as the switching instant is in the observation window. Thus, a maximum delay equals to the length of the observation window exists. Prior knowledge on the process such as "prohibited" switching sequences or minimal time between two consecutive switchings, can also help to limit the number of generated residuals or paths to be considered.

Path discernability

In what follows, we are interested in the conditions guaranteeing the discernability of the various paths enumerated on an observation window. These conditions ensure the uniqueness of the recovered active path µ * during the path recognition process.

Definition 4 (Path discernability) Two paths µ 1 ∈ Θ h and µ 2 ∈ Θ h are discernible on an observation window [kh, k] if their respective corresponding residuals r µ 1 ,h (•) and r µ 2 ,h (•) are not simultaneously null when one of the two paths is active on the considered observation window.

In order to establish the discernability conditions of two different paths, let us consider two paths µ 1 ∈ Θ h and µ 2 ∈ Θ h on an observation window

[k -h, k]. We denote Y µ 1 k-h,k (respectively Y µ 2 k-h,k
) the output vector related to the system when it undergoes the path µ 1 (respectively µ2 ). We suppose that at an instant k, the active path on the observation window is the path µ 1 . This information being unknown, we have to analyze the possibilities that the path µ 1 or the path µ 2 are in adequacy with the system's data. Using equation ( 5), the expressions of the residuals r µ 1 ,h (•) and r µ 2 ,h (•), when µ 1 is the active path, are given by:

r µ 1 ,h (k) = 0 r µ 2 ,h (k) = Ω µ 2 ,h Y µ 1 k-h,k -T µ 2 ,h U k-h,k (7)
Adding and taking away Y µ 2 k-h,k from the expression of r µ 2 ,h (•), one obtains:

r µ 1 ,h (k) = 0 r µ 2 ,h (k) = Ω µ 2 ,h Y µ 1 k-h,k -Y µ 2 k-h,k + Y µ 2 k-h,k -T µ 2 ,h U k-h,k (8)
As by definition

Ω µ 2 ,h Y µ 2 k-h,k -T µ 2 ,h U k-h,k = 0, one has: r µ 1 ,h (k) = 0 r µ 2 ,h (k) = Ω µ 2 ,h Y µ 1 k-h,k -Y µ 2 k-h,k (9)
Equation ( 9) clearly points out that the residual calculated for the path µ 2 (nonactive path) directly depends on the difference between the system's outputs when the mode sequence evolves according to the two paths µ 1 and µ 2 , the system being excited by the same inputs in both cases. Therefore, a necessary and sufficient condition for the residuals r µ 1 ,h (k) and r µ 2 ,h (k) not to be simultaneously equal to zero is:

Y µ 1 k-h,k -Y µ 2 k-h,k / ∈ N r (Ω µ 2 ,h ) (10)
where N r stands for the operator "right null space". Condition (10) has to be analyzed in order to deduce the discernability conditions. According to (3), one has:

Y µ 1 k-h,k -Y µ 2 k-h,k = O µ 1 ,h -O µ 2 ,h x(k -h) + T µ 1 ,h -T µ 2 ,h U k-h,k (11)
where x(kh) is the value of the system's state at the initial instant of the observation window.

From (11), we obtain:

Ω µ 2 ,h (Y µ 1 k-h,k -Y µ 2 k-h,k ) = Ω µ 2 ,h O µ 1 ,h x(k -h) + Ω µ 2 ,h T µ 1 ,h -T µ 2 ,h U k-h,k (12) If Y µ 1 k-h,k -Y µ 2 k-h,k
belongs to the right null space of Ω µ 2 ,h , one has:

Ω µ 2 ,h O µ 1 ,h x(k -h) + Ω µ 2 ,h T µ 1 ,h -T µ 2 ,h U k-h,k = 0 ( 13
)
The relation is satisfied "for almost every initial state" 2 x(kh) if the following necessary and sufficient condition is satisfied:

Ω µ 2 ,h O µ 1 ,h = 0 Ω µ 2 ,h T µ 1 ,h -T µ 2 ,h U k-h,k = 0 (14)
Therefore, the paths µ 1 and µ 2 are not discernible on a time window [kh, k] if the relations (14) are satisfied.

Theorem 2 (Path discernability) Two paths µ 1 and µ 2 of a switching system are discernible on an observation window [kh, k], "for almost every initial state" x(kh), if:

Ω µ i ,h O µ j ,h = 0, i, j ∈ {1, 2} , i = j (15)
or

Ω µ i ,h T µ j ,h -T µ i ,h U k-h,k = 0 i, j ∈ {1, 2} , i = j (16)
where U k-h,k is the vector containing the system's input stacked on the observation window.

The proof of this theorem directly comes from the preceding remarks.

Remark 1 (Dependency to the initial state) In theorem 2, the expression "for almost every initial state" holds owing to the fact that the discernability of the paths cannot be ensured for any initial state x(k-h). In fact, for certain particular values of x(k-h), the relation ( 13) is always satisfied independently of the input sequence U k-h,k . For example, in the situation where O µ 1 ,h has full rank, for 13) is satisfied for every input sequence U k-h,k , where Φ belongs to the right null space of Ω µ 2 ,h and O µ 1 ,h † is the pseudo-inverse of O µ 1 ,h .

x(k-h) = O µ 1 ,h † Φ -T µ 1 ,h -T µ 2 ,h U k-h,k , equation (

Academic example

We present here an academic example of a switching system characterized by three modes and the matrices of the models describing the different modes are:

A 1 = -0.211 0 0 0.521 A 2 = 0.691 0 0 -0.310 A 3 = 0.153 0 0 0.410 B = 2 -1 T C = 1 2 (17)
Figure 1 shows the input u(•), the output y(•), the state x(•) the mode sequence µ (•) . The vertical dashed lines on the third graphic of figure 1 mark the time instants where switchings occur. The fourth graphic plots the mode sequence described by the mode's selection variable µ (•) .

As Ω k,h l i O k,h l j = 0, µ i , µ j ∈ Θ 2 , µ i = µ j , Θ 2 being the set of all paths of length 2, the condition (15) of theorem 2 is respected. Condition ( 16) is tested at every moment. If it is not satisfied, no decision is taken concerning the recognition of the active path. In order to perform the determination of the active path at every moment from the system's input and output signals, we consider an observation window of length 3, leading to the consideration of the set Θ 2 of all paths of length 2 on the observation window. Figure 2 presents the evolution of the calculated residuals. The different graphics on the figure show the residuals r (i • j),h (•), i, j ∈ {1, 2, 3} corresponding to the paths of length 2. Only one residual equals zero at each instant, the index (i • j) of this residual corresponding to the active path on the considered time window.

As explained in section 4, to reduce the number of residuals to be analyzed during the active mode determination process, it is possible to only consider the paths describing the mode sequence when the system remains in the same mode all over the duration of the observation

Conclusion

This paper proposed a method for the determination of the active mode and the switching instants of a switching system, using only the system's input and output data. Discernability conditions have also been examined. A point to be deepened is the situation where all the modes of the system are not previously indexed. In this case, one does not have a complete knowledge of all the operating regimes of the system. Therefore, it is necessary to simultaneously proceed to the detection of not indexed modes and to the estimation of their parameters.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Input u(•), output y(•), state x(•), mode sequence µ (•)

In fact, the existence of the projection matrix is directly linked to the observability of the system and to the length of the observation window[START_REF] Gertler | Fault detection and isolation using parity relations[END_REF]

see remark 1 for the explanation of the expression "for almost every initial state"


Yvon Tharrault
email: yvon.tharrault@ensem.inpl-nancy.fr

Gilles Mourot

Didier MAQUIN Jos É Ragot

Fault detection and isolation with robust principal component analysis

Keywords: Principal component analysis, robustness, fault signature, fault detection and isolation, outliers

Principal component analysis (PCA) is a powerful fault detection technique which has been widely used in process industries. However, a main drawback of PCA is that it is based on least squares estimation techniques and hence fails to account for outliers which are common in physical processes. This paper is concerned with the fault detection and isolation problem. The proposed method does not require a data matrix without outliers for a PCA model design. Indeed, the approach directly uses the eventually corrupt database to elaborate a robust PCA model allowing fault detection. Then reconstruction principle and fault signatures analysis are used for fault isolation.

Introduction

Principal component analysis (PCA) is widely used as a multivariate statistical method for fault detection, isolation and diagnosis. PCA is mainly based on the description of linear relations between variables and optimises a MSE (Mean Square Error) criterion. It is wellknown that the estimation based on a criterion like MSE is less robust to outliers than that resulting from other criteria like error absolute value [START_REF] Hubert | Robpca: a new approach to robust principal component analysis[END_REF]. Let us recall that the traditional approach of the PCA uses a preliminary calculation of the average of data and their covariance matrix; average and variance are sensitive to outliers, and the obtained results are often not exploitable because too biased by the influence of these outliers. To take outliers into account, a robust covariance matrix of the data can be used to construct a robust principal component analysis. For that, [START_REF] Croux | Principal component analysis based on robust estimators of the covariance or correlation matrix: Influence functions and efficiencies[END_REF] built particular functions of influence and the asymptotic variances which result from them. [START_REF] Engelen | A comparison of three procedures for robust pca in high dimensions[END_REF] proposed the robust approach ROBPCA, which combines revealing projections with a robust estimate of the variance matrix. This technique produces estimates which appear robust in the presence of outliers. [START_REF] Brown | Adaptive m-estimators for robust covariance estimation[END_REF] focused on the robust estimate of the covariance matrix for multidimensional systems. Other approaches dealing with the problem of robustness were proposed in [START_REF] Böhning | A note on the maximum deviation of the scalecontaminated normal to the best distribution[END_REF] by using a scale-contaminated distribution law and in (Salibian- [START_REF] Salibian-Barrera | Pca based on multivariate mm-estimators with fast and robust bootstrap[END_REF] where the authors develop an approach based on a moment calculation.

Our presentation is devoted to the problem of fault detection and isolation in data. In general, faults result from process dysfunctions or from the system of measurement acquisition. The contribution essentially deals with the detection and isolation of outliers by using complementary tools: robust principal component analysis, data reconstruction and residual analysis with the fault signatures. Section 2 is a short reminder, on one hand, of the principal component analysis in the traditional case and, on the other hand, of the robust principal component analysis. A detection and isolation procedure for outliers is proposed in section 3, then, in section 4, is applied to an example of synthesis emphasizing the generation of fault signatures.

PCA fault detection and isolation

Let us consider a data matrix X ∈ ℜ N ×n , with vector lines x T i , which gather N measurements collected on the n system variables.

Traditional approach

In the traditional PCA case, data are supposed to be collected on a system being in a normal process operation. PCA determines an optimal linear transformation of the data matrix X in terms of capturing the variation in the data:

T = XP et X = T P T (1)
with T ∈ ℜ N ×m the principal component matrix and the matrix P ∈ ℜ m×m the one that contains the principal vectors which are the eigenvectors associated to the eigenvalues λ i of the covariance matrix (or correlation matrix) Σ of X:

Σ = P ΛP T avec P P T = P T P = I m (2)
where Λ = diag(λ 1 . . . λ m ) is a diagonal matrix with diagonal elements in decreasing magnitude order. The relations (1) are meaningful when the dimension of the representation space is reduced.

Once the component number ℓ to retain is determined, the data matrix X can be approximated. For that, the eigenvectors matrix is partitioned into the form:

P = P P P ∈ R n×ℓ (3)
From the decomposition (1), X is the principal part of the data explained by the ℓ first eigenvectors and the residual part X is explained by the remaining components:

X = X P P T = XC ℓ (4) E = X -X = X(I -C ℓ ) (5)
where the matrix C ℓ = P P T is not equal to the identity matrix.

Robust approach

A major difficulty of PCA comes from its sensitivity to outliers. In order to reduce this sensitivity, various techniques are usable and in particular that which consists in carrying out PCA directly on the data possibly contaminated by outliers. An alternative is to seek principal directions robust to these outliers. [START_REF] Fekri | Robust weighted orthogonal regression in the errorsin-variables model[END_REF] define a "local" matrix of variance in the sense that the suggested form tends to emphasize the contribution of close observations in comparison with distant observations (outliers). The matrix is defined in the following way according to the observations x i :

V = N -1 i=1 N j=i+1 w i,j (x i -x j )(x i -x j ) T N -1 i=1 N j=i+1 w i,j (6)
where the weights w i,j themselves are defined by:

w i,j = exp - β 2 (x i -x j ) T Σ -1 (x i -x j ) (7)
β being a turning parameter to reduce the influence of the observations faraway, the authors recommend a value close to 2. Thanks to the presence of adapted weights w i,j , PCA can then be carried out on this "new" matrix of covariance considered robust with respect to outliers.

Fault detection

Data reconstruction

PCA can be used for fault detection, these faults resulting in outliers which are highlighted by projection onto the residual space [START_REF] Dunia | A subspace approach to multidimensional fault identification and reconstruction[END_REF]. The PCA model being known, a new measurement vector x can be decomposed as below:

x = x + x , x = C ℓ x , x = (I -C ℓ ) x (8)
where x and x are respectively the projections of x on the principal space and residual space. From (8), it is possible to estimate a particular component of the vector x, for example the R th , where R is a subset containing the indices of the reconstructed variables. However, the presence of outliers in the observation vector x returns the estimated x sensitive to this value. It is then preferable to express this estimated x by using only the fault-free part of the observation vector x.

The reconstruction [START_REF] Dunia | A subspace approach to multidimensional fault identification and reconstruction[END_REF] of process faults consists in estimating the reconstructed vector xR by eliminating the effect of the faults. Matrix Ξ R indicates the reconstruction directions. This matrix is orthonormal with dimension (n × number of reconstructed variables) and is constructed with 0 and 1, where 1 indicates the reconstructed variables from the other variables (with 0), for example Ξ R = [0 1 0 1 0] T for n = 5 and one reconstructed variable (R = {2, 4}).

The expression for the reconstruction xR of the variable x is given by [START_REF] Dunia | A subspace approach to multidimensional fault identification and reconstruction[END_REF]:

xR = [I -Ξ R ( ΞT R ΞR ) -1 ΞT R )]x (9)
where

Ξ = (I -C ℓ )Ξ R .
Let us note that if ( ΞT R ΞR ) -1 exists the R th variable is completely reconstructable or else the fault is partially reconstructable. This two cases are presented by [START_REF] Dunia | A subspace approach to multidimensional fault identification and reconstruction[END_REF]. In the following, only completely reconstructable faults are considering.

Residual generation

In a diagnosis objective, residuals are generated for fault detection and isolation. The reconstruction procedure is successively applied to all the components of x. The reconstructions obtained (9) are then compared with the measurements. The residuals are obtained by projecting the reconstructed variables onto the residual space. Residuals are defined by xR , projection of xR onto the residual space:

xR = P (ℓ) R x (10)
P (ℓ) R = (I -C ℓ ) -ΞR ( ΞT R ΞR ) -1 ΞT R (11) Remark 1. Matrix P (ℓ)
R has the following proprieties:

P (ℓ) R Ξ R = 0 and Ξ T R P (ℓ) R = 0 (12)
It means that the components of xR are not sensitive to the R th components of x. This remark can be used to identify which component of x is disturbed by faults.

For example, considering a measurement x composed with the true value x * , a noise ǫ with null mean and one fault of amplitude d and direction Ξ F , where F is a subset containing the indices of the reconstructed variables:

x = x * + ǫ + Ξ F d (13)
then the residual is:

xR = P (ℓ) R (x * + ǫ + Ξ F d) = P (ℓ) R (ǫ + Ξ F d) (14)
and its expected value is:

E(x R ) = P (ℓ) R Ξ F d (15)
-if the reconstruction direction Ξ R is the same as the fault, i.e. if R = F ,then all components of the vector P (ℓ) R Ξ F are null and E(x R ) = 0 -if the reconstruction direction Ξ R is different from the fault direction, then all components of the vector P (ℓ) R Ξ F are a priori not null except the R th components. Then, the analysis of the residual amplitudes xR for all possible combinations shows the presence of faults and makes it possible to determine the components of the measurement affected by this fault.

Numerical example

4.1. Numerical example -mono-fault case Data generation A simple example based on four variables (x 1 , x 2 , x 3 et x 4 ) and two models is used. The data matrix X includes N = 240 measurements defined in the following way:

x i,1 = v 2 i + 1 + sin(0.1i), v i ∼ N (0, 1) (16) x i,2 = x i,1 , x i,3 = -2x i,1 , x i,4 ∼ N (0, 1)
Realizations of centered normal distributions with the same standard deviation equal to 0.02 are added to these four variables. The variable x 4 , independent of other variables, is a perturbation for PCA. A constant bias of amplitude equal to 3 simulates the presence of outliers δx 1 , δx 2 , δx 3 affecting the variables x 1 , x 2 and x 3 : from 24 to 44 for the variable x 1 , from 80 to 100 for the variable x 2 , from 140 to 160 for the variable x 3 . It's important to notice that 60 observations contained abnormal values, hence 25 percent of the data are contaminated by these values. The objective is to detect and especially isolate them.

Sensitivity analysis and theoretical fault signature

The data in the table 1 summarize the relationship between residual sensitivity xR (10) and outliers or faults δx 1 , δx 2 and δx 3 (fault δx 4 on variable x 4 is not considered). This table was constructed while taking into account proprieties of the matrix P (ℓ) R (12). For example, the first four residuals x11 to x14 (relative to variables x 1 , x 2 , x 3 and x 4 ) were obtained by projection onto the residual space of reconstructed variables without variable x 1 . As the first line and the first column of P ℓ 1 are null, according to (12), the residual x11 is not sensitive to variables x 1 , x 2 and x 3 and consequently to potential faults δx 1 , δx 2 or δx 3 affecting these variables. Moreover, the residuals x12 , x13 and x14 are not sensitive to variable x 1 and thus to the fault δx 1 which can affect them. To summarize these different situations, the symbols × and 0 translate, or not, the fault influence on the residuals. The other parts of the table were constructed with this same principle by considering the different projection matrices P . By analysing the dependence of the columns of the signature matrix, one can establish necessary conditions allowing the fault detection and isolation. Let us note that only two projection matrices and two residuals are necessary for fault detection and isolation. For example, matrices P (ℓ) 1 and P (ℓ) 2 (11), allow to build the residuals x12 (relative to x 2 ), x21 (relative to x 1 ) which, permit to detect and isolate one of the three faults. Indeed, table (1) indicates that with these two residuals, the signature faults δx 1 , δx 2 and δx 3 are respectively (0 ×), (× 0) and (× ×); these three signatures are independent and thus the faults are isolable from each other.

Table 1. Fault signatures r = 1 r = 2 r = 3 r = 4 x11 x12 x13 x14 x21 x22 x23 x24 x31 x32 x33 x34 x41 x42 x43 x44 δx 1 0 0 0 0 × 0 × × × × 0 × × × × 0 δx 2 0 × × × 0 0 0 0 × × 0 × × × × 0 δx 3 0 × × × × 0 × × 0 0 0 0 × × × 0

Fault detection

Through the use of raw data, we established a robust PCA model by applying the propositions of section 3. The analysis of the decrease of the standardized eigenvalues of the covariance matrix (85.94, 13.99, 0.04, 0.03), allows to retain two principal components (ℓ = 2). As indicated by remark 1, in the case of the residual generated without using variable x 1 , only faults affecting the variables x 2 and x 3 are detectable on the residuals x12 , x13 and x14 . Graphics of the figure 1 are relative to a global indicator ∆ R (norm of the projection vector weighted by the covariance matrix

V R = P (ℓ) R ΣP (ℓ)T R of these projections) computed for each observation: ∆ R = xR 2 V -1 R (17)
A simple jump test on the ∆ R quantity (like Page-Hinkley for example) allows to determine the index of the fault observations. Detection and isolation are realised without ambiguity and are in accordance with the theoretical results of the isolation procedure (table 1). We can notice that a traditional non-robust PCA gives no significant result for fault detection.

Numerical example -multi-fault case

Data generation

The matrix X includes N = 108 observations of a vector x with 8 components generated in the following way:

x i,1 = v 2 i + sin(0.1i), x i,2 = 2 sin(i/6) cos(i/4) exp(-i/N ), v i ∼ N (0, 1) (18) x i,3 = log(x 2 i,2 ), x i,4 = x i,1 + x i,2 , x i,5 = x i,1 -x i,2 x i,6 = 2x i,1 + x i,2 , x i,7 = x i,1 + x i,3 , x i,8 ∼ N (0, 1)
On the data thus generated were superimposed realizations of random variables with centered normal distribution and standard deviations equal to 0.02 as well as faults δx 1 , δx 2 , δx 3 , δx 4 represented by a bias of amplitude equal to 3 and defined in the following way: observations from 10 to 24 for the variable x 1 , observations from 35 to 49 for the variables x 2 and x 3 , observations from 60 to 74 for the variables x 3 and x 4 , observations from 85 to 99 for the variable x 4 . In the following, these four intervals are indicated by I 1 , I 2 , I 3 , I 4 .

Sensitivity analysis

Concerning the a priori analysis of fault isolation, we limit ourselves to giving a reduced table of signatures (table 2) established from the properties (12). It reveals only some possible faults, noted δ in the first line, those affecting variables 1, 2, 3, 4 and those affecting the couples of variables {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}. The first column relates to the norm ∆ R of the residual vectors obtained by reconstruction-projection of the variables by using all the components of x except that with the indices R. The residuals are defined by (10). This table, which the reader will be able to extend , provides a correspondence be-

∆ 1 ∆ 2 ∆ 3 ∆ 4 ∆ 5 ∆ 6 ∆ 12 ∆ 13 ∆ 14 ∆ 15 ∆ 16 ∆ 23 ∆ 24 ∆ 25 ∆ 26 ∆ 34 ∆ 35 ∆ 36 δ 1 0 × × × × × 0 0 0 0 0 × × × × × × × δ 2 × 0 × × × × 0 × × × × 0 0 0 0 × × × δ 3 × × 0 × × × × 0 × × × 0 × × × 0 0 0 δ 4 × × × 0 × × × × 0 × × × 0 × × 0 × × δ 12 × × × × × × 0 × × × × × × × × × × × δ 13 × × × × × × × 0 × × × × × × × × × × δ 14 × × × × × × × × 0 × × × × × × × × × δ 23 × × × × × × × × × × × 0 × × × × × × δ 24 × × × × × × × × × × × × 0 × × × × × δ 34 × × × × × × × × × × × × × × × 0 × ×
tween the symptoms ∆ R and the faults δ R . For example, the defect δ 2 affects all projections except those established without components 2, {1, 2}, {2, 3}, {2, 4}, {2, 5}, {2, 6}. Let us analyse the figure 2. The variable 1, biased for the observations of the interval I 1 , is not used for the reconstruction and the other variables which are used for the reconstruction do not present any bias. For these observations, the reconstructions are thus correct, emphasizing the first graph (starting from the top of the figure) which shows the superposition of the reconstructed variables (symbol 'o') with the true variables (in practice the latter are unknown, but at this stage where the data are generated, the comparison is possible). The measurement of the variable is also indicative (continuous line) in order to compare it with the reconstruction. This result is confirmed by the last graph of the figure 2 where the norm of the vector projection (17) was traced. For the observations of the interval I 1 this norm is close to the value 0 thus testifying to the absence of outliers in the variables used for the reconstruction and projection, i.e. all the variables except x 1 . Let us note that the three other groups of observations (I 2 , I 3 , I 4 ) are affected by faults, without knowing exactly which components of the measurement vector are faulty. Finally, by taking into account the fault presence in the four intervals, the examination of the figure 2 concludes that: -in each interval I 2 , I 3 , I 4 , a variable other than x 1 is faulty Other projections (not presented here) are built and are interpreted in a similar way. The table 3 summarizes the conclusions resulting from the projection analysis. The line ∆ 1 relates to the reconstructed residuals without using the first variable, the symbol 0 attests the fault absence in the considered interval. The diagnosis is then:

-in the interval I 1 , x 1 is faulty -in the interval I 2 , x 2 and x 3 are faulty -in the interval I 3 , x 3 and x 4 are fautly -in the interval I 4 , x 4 is faulty

I 1 I 2 I 3 I 4 ∆ 1 0 × × × ∆ 23 × 0 × × ∆ 24 × × × 0 ∆ 34 × × 0 0

Conclusion

Simulation results confirm that for data not contaminated by errors, classical PCA and robust PCA give similar results. In the other situations where outliers corrupt the data, traditional PCA proves to be ineffective, whereas its robust version gives completely satisfactory results. On the treated examples, the presence of approximately 25 percent of outliers authorizes a correct estimation of the principal directions, then the estimation is not very sensitive to these values. A PCA model can thus be built directly from the available data containing potential faults.

The most important result concerns the diagnosis of the systems, applied here to the detection and isolation of outliers. For that, we showed how to build fault indicators. The use of the principle of reconstruction and projection of the reconstructed data together made it possible to detect and isolate outliers in an effective way. The procedure suggested here, is not limited in theory by the number of variables. However, the computational load is likely to become incompatible with an on-line treatment of the data and a reduction of the reconstruction and projection number is possible; this point will require a detailed attention in the follow-up of our work.

Fig. 1 .

 1 Fig. 1. Global indicator for x1 and x2

Fig. 2 .

 2 Fig. 2. Variables reconstruction without using variable 1 and global indicator

Table 2 .

 2 Reduced table of faults signature

Table 3 .

 3 Fault signatures

Fault detection

From the contaminated data, the robust PCA model, with four principal axes, was chosen. Without carrying out the reconstruction, the observations were projected onto the residual space. The analysis of the residual norm thus generated by using all the variables reveals the presence of defects in the four intervals I 1 , I 2 , I 3 , I 4 , without being able to call into question a particular variable. This phase of detection is now supplemented by a stage of fault isolation. The reconstruction is then carried out from all the variables except the variable 1, then starting from all the variables except the variables 1 and 2,... the last reconstruction being made from all the variables except the variables 7 and 8. The figures 2 visualize the reconstructions of variables without using the variable 1. This figure shows the reconstruction of the first seven variables which are to be associated with the column ∆ 1 of table 2 specifying the isolable faults. The N reconstructed data were then projected onto the residual space. For each observation fault indicator ∆ R (17) were calculated.


Etienne Dague
email: edague@laas.fr

Atomic Force Microscopy to explore Electroporation effects on cells

Keywords: Atomic Force Microscopy, Electroporation, Elasticity, Cell topography, cytoskeleton, bacteria, adhesion

Atomic Force Microscopy (AFM) is more and more used in life science. Its ability to provide images of living cells as well as mechanical or adhesion maps makes it a technology that cannot be ignored. In the context of electroporation (EP) which undoubtly affects the cells membrane or wall, a technology able to probe the cell surface is more than interesting. This chapter describes the principle of the AFM technology and especially the latest multiparametric imaging modes developed recently. It then demonstrates that AFM can be used to probe cell's morphology modifications induced by electric pulses. We then show that EP modifies cell's nanomechanical properties and that the actin cytoskeleton plays a major role in this process. Finally we shed light on the effects of EP on bacteria as probed by AFM.

In this latest example it must be noticed that no mechanical modifications are induced, but the adhesion properties of the bacteria are dramatically reduced by Pulsed Eelectric Fields (PEF).

Altogether the chapter shows the interest of applying AFM on cells exposed to EP, in order to get a better fundamental understanding of EP effect on cells or bacteria.

Introduction

Electroporation (EP) consists in applying electric fields of various intensity and/or duration on cells in order to modify their interface and let non-permeant molecules go through the plasmic membrane [START_REF] Wong | Electric field mediated gene transfer[END_REF] . EP has proven to be efficient to potentiate cytotoxic molecules (electrochemotherapy in cancer treatment) and has the potential to be used in gene transfer [START_REF] Testori | Utility of electrochemotherapy in melanoma treatment[END_REF][START_REF] Bodles-Brakhop | Electroporation for the Delivery of DNA-based Vaccines and Immunotherapeutics: Current Clinical Developments[END_REF] .

However the mechanisms of cell membrane modifications are poorly understood and limit the applications of this technology [START_REF] Teissie | Mechanisms of cell membrane electropermeabilization: A minireview of our present (lack of ?) knowledge[END_REF] . Exploring and understanding the modifications induced by EP on cell's membrane is indeed a challenge that has been addressed by measuring the entry of fluorescent or radioactive molecules in pulsed cells or by transmembrane potential measurements. However using these techniques, the direct physical effect on the membrane is not obvious and the word electroporation itself is questionable as no pores were ever reported.

Electropermeabilization is therefore sometimes preferred. The cell membrane permeability is, indeed, modified and understanding electroporation effects is therefore a matter of cell membrane exploration. In this context the atomic force microscope is a pertinent technology as it makes it possible to analyze cells surface in their native environment, in this case before and after electric pulses application. Atomic Force Microscopy (AFM) is an imaging technology developed in the 80 th by physicists [START_REF] Binnig | Atomic force microscope[END_REF][START_REF] Hansma | Scanning tunneling microscopoy and atomic force microscopy : application to biology and technology[END_REF][START_REF] Gerber | How the doors to the nanoworld were opened[END_REF] . It has been more and more applied on biological samples (virus, bacteria, mammalian cells, DNA, proteins…) since the 90 th8 and recently to explore the effects of EP on cells morphology and nanomechanical properties [START_REF] Chopinet | Destabilization induced by electropermeabilization analyzed by atomic force microscopy[END_REF][START_REF] Chopinet | Imaging living cells surface and quantifying its properties at high resolution using AFM in QI TM mode[END_REF][START_REF] Louise | AFM sensing cortical actin cytoskeleton destabilization during plasma membrane electropermeabilization[END_REF][START_REF] Pillet | Cell wall as a target for bacteria inactivation by pulsed electric fields[END_REF][START_REF] Thompson | Disruption of the actin cortex contributes to susceptibility of mammalian cells to nanosecond pulsed electric fields[END_REF] .

But more than an imaging technology AFM is also a force machine able to measure forces in the pN range [START_REF] Neuman | Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy[END_REF] . This capability can be used to measure the nanomechanical properties of cells and especially after EP.

In this chapter I will describe the AFM technology and present its application to better understand EP.

Atomic Force microscopy

The AFM principle [START_REF] Binnig | Atomic force microscope[END_REF] relies on the measure of interaction forces between a sharp tip and the analyzed sample. The tip is scanned over the sample, or conversely, the sample is scanned over the tip, thanks to a piezo electric ceramic that can be moved in the 3 spatial directions: x, y and z. In order to allow a displacement of the tip in the z direction, it is mounted on a cantilever, which deflection is monitored via an optical device based on a laser and a 4 quadrants photodiode. The resolution is therefore directly linked to the tip size. The sharpest tip, like for example carbon nanotubes, will achieve the highest resolution. The drawback is that such tips are both stiff and delicate. As a consequence they are not adapted for living cells and tips presenting pyramidal geometry (curvature radius around 20 nm) are preferred for living cells exploration. Another parameter that can limit AFM resolution is the scanning velocity. Images, or force maps are recorded line-by-line, or point-by-point and this is a time consuming process. A 512 lines image recorded at a scan rate of 1Hz requires 8 minutes and 30 seconds (each line is scanned 2 times). A solution to increase the resolution can be to decrease the scanned area. Classical piezo can scan areas up to 100x100 µm [START_REF] Testori | Utility of electrochemotherapy in melanoma treatment[END_REF] . By decreasing the analyzed surface, the resolution is mechanically increased. However this has a limit linked to the tip size. In a 10x10 µm 2 picture made of 512 lines, the thickness of each line is of 19 nm, what is close to the typical curvature radius of tips used for living cells. In this case, decreasing the scan size will not result in increasing the resolution so much, each line being smaller than the tip size.

The main advantage of AFM in life science is its ability to work in liquids, like growing media; at a controlled temperature, i.e. 37°C; under gas flow, typically CO2 5%. In these conditions living cells can be observed during several hours, in their native condition and/or after a treatment (drug, EP, temperature modification…).

Imaging

AFM images represent the topography of the sample, which is deduced from the cantilever deflection. In contact mode (figure 1), the tip is applying a constant and arbitrary force on the sample during scanning. The force is kept constant by modulating the tip or sample height trough a feedback loop. This mode is easy to use and provides high quality images. However the tip, while scanning, induces lateral forces that can damage the sample, especially in the case of a soft sample like a cell. Conversely the lateral forces induce a friction, which is recorded as a lateral deflection of the tip. This lateral deflection is material dependent and can therefore be used to decipher between materials under investigation. The drawback of these lateral forces is that the biological sample has to be firmly immobilized on the surface to overcome them (figure 1). Thus strategies were developed to perform non denaturing immobilization [START_REF] El Kirat | Sample preparation procedures for biological atomic force microscopy[END_REF][START_REF] Touhami | Nanoscale Mapping of the Elasticity of Microbial Cells by Atomic Force Microscopy[END_REF][START_REF] Dague | Assembly of live micro-organisms on microstructured PDMS stamps by convective/capillary deposition for AFM bio-experiments[END_REF][START_REF] Formosa | Generation of living cell arrays for atomic force microscopy studies[END_REF] . It must be noticed that mammalian cells are usually spreading on their support and that no special immobilization procedure is needed (figure 1). However, these cells are very soft (Young modulus (YM) are ranging from 1 to 100 kPa) and are usually difficult to image in contact mode.

To overcome this limitation the historical AFM manufacturer developed a non-contact mode named Tapping® mode. Here the tip is oscillated near to its resonance frequency, and the oscillation amplitude is kept constant during scanning. The phase shift can also be interpreted as material dependent, and the phase signal used to decipher between different materials. 12. Force Spectroscopy: nanomechanics, adhesion.

As already stated, the AFM is a force machine. In the force spectroscopy mode, the tip is no longer scanned in the x,y direction but in the z direction. While approaching the sample, the tip is first encountering no forces, then it contacts the surface and finally the cantilever and the sample (if it is soft) are deflected. The approach force curve can be converted into an indentation force curve. From the indentation of the tip in the sample, the mechanical properties can be deduced, using models like the Hertz model. In this model, as in many others, the applied force is a function of the sample elasticity or Young Modulus. A higher value of E means that a higher force is required to achieve the same indentation in the sample.

Thus the link between elasticity and the Young modulus is biologically unintuitive: high Young modulus do not mean easily deformable. The Young modulus is a pressure measured in Pa (N.m -2 ). The exact absolute value of a sample Young modulus is a complex question. The sample anisotropy, the shape of the tip sample contact, the dispersed energy, the sample viscosity, etc… should be taken into account to achieve an exact measure. For this reason, Young modulus absolute values should be used with caution. Nevertheless, the comparison of Young modulus values measured on the same sample, but in different conditions is valuable and provides interesting insight into the nanomechanical modifications induced by a treatment.

During the retraction process, the cantilever comes back to zero deflection, but if an interaction has occurred between the tip and the sample, a higher force will be needed to detach the tip from the sample. The adhesion forces can be due to electrostatic or hydrophobic interactions [START_REF] Ahimou | Probing microbial cell surface charges by atomic force microscopy[END_REF][START_REF] Dague | Chemical force microscopy of single live cells[END_REF] or to specific biological interactions between a ligand and a receptor for example [START_REF] Hinterdorfer | Detection and localization of individual antibody-antigen recognition events by atomic force microscopy[END_REF][START_REF] Hinterdorfer | Detection and localization of single molecular recognition events using atomic force microscopy[END_REF] . Measuring specific forces require to functionalize the tip, which can be performed through the use of linkers grafted on the AFM tip.

In the Force Volume (FV) mode (Figure 1), force curves are recorded according to a matrix of point. Each force curve is recorded and their analysis leads to adhesion and/or elasticity maps.

Multiparametric modes

Multiparametric modes [START_REF] Dufrêne | Multiparametric imaging of biological systems by force-distance curve-based AFM[END_REF] (figure 1) were developed recently by two AFM manufacturers.

They are optimizing the FV by increasing the number of force curves recorded in the same time. The resolution is therefore increased and/or the acquisition time is reduced. In this new mode, and thanks to an increased resolution, both, valuable topography, adhesion and elasticity maps are recorded at the same time. Moreover, multiparametric modes are convenient to image soft and loosely immobilized biological samples. Indeed, as the tip is retracted between each force curves, thus no or low lateral forces are exerted on the sample.

The main limitation remaining in these modes is the z piezo course. It must be large enough to detach the tip from the cell; however this inconvenient can be avoided by a selecting the appropriate tip (shape) and cantilever (spring constant)

Imaging effects of EP on cells

In order to explore EP effects on mammalian cells shape, Chopinet et al. [START_REF] Chopinet | Destabilization induced by electropermeabilization analyzed by atomic force microscopy[END_REF] , in 2013, first imaged, by AFM, cells immediately fixed after EP. No specific morphological alterations were detected neither in front of the electrode nor on regions perpendicular to the electrodes.

Globally the measured elasticity is decreased after EP (from 30.0 +/-2.0 kPa to 20.5 +/-1.0 kPa), but no link with the electrodes position appears. Then living cells were imaged across time. Figure 2A presents the same living cell imaged during 50 minutes and figure 2C cross sections of the cell measured after 9, 24 and 50 minutes of imaging. These topographical images were recorded in the multiparametric quantitative imaging mode. It demonstrates that the cells are not changing their shape during a 50 minutes period of time. At higher resolution (see insets in the down right corners and supplementary material in [START_REF] Chopinet | Destabilization induced by electropermeabilization analyzed by atomic force microscopy[END_REF] ), images show that the membrane is smooth, homogeneous and cytoskeleton fibers are detected. The membrane and cytoskeleton remain stable during all the experiment. In a second step, figure 2B, EP (8 square-wave electric pulses of 5 ms duration at 400 V/cm) was applied at a frequency of 1 Hz through stainless steel parallel electrodes directly on the Petri dish in buffer after the first image of the cell. The next 2 images and the cross sections presented figure 2D (black arrow) demonstrate a swelling of the cell induced by EP. At higher resolution, cell membrane modifications appear to be reversible. After 18 minutes: i) many extensions rippling membrane can be seen and ii) no cytoskeleton fibers are observed any more. At the end of the experiment (50 minutes) the cell membrane seems to recover, looking like before EP.

EP is also used in food industry to eradicate undesirable bacteria. Pillet et al. [START_REF] Pillet | Cell wall as a target for bacteria inactivation by pulsed electric fields[END_REF] , in 2016 explored the effect of EP (5 μ s-pulses at frequency of 1 kHz in a 4.1 mM NaCl solution, with a conductivity of 500μ S/cm and a pH of 7), on Bacillus pumilus cell surface and shape. They both analyzed vegetative bacteria and spores. Figure 3 a, b, c andd d, f, andh respectively on images a, c, e andg. The color scale, representing the highness, is given on the right of each image. Adapted from [START_REF] Pillet | Cell wall as a target for bacteria inactivation by pulsed electric fields[END_REF] The surface roughness has been determined to be 1.4 +/-0.5 nm before EP and 6.1 +/-5.2 after EP. Similarly to mammalian cells, EP induces a swelling of vegetative bacteria: the mean cell volume is 1.9 +/-0.7 µm 3 before EP and increases to 3.0 +/-1.6 µm 3 after pulses application. Vegetative bacteria have thus a smooth surface that is dramatically impaired by EP. The effect is emphasized on higher resolution images (figure 3b andd).

As far as spores are concerned (figure 3 e, f, g, h), their surface structure is also disturbed by EP. Bacillus spores are covered by coat proteins forming parallel nanostructured named ridges (see doted lines in figure 3 f). These structures are clearly affected by EP (figure 3 h) but no impact on roughness or cell volume were quantified (roughness was 5.5 +/-2.9 nm for untreated spores, 4.2 +/-1.7 nm after EP exposure and volume was 0.9 +/-0.2 μm 3 for untreated cells and 0.9 +/-0.3 μm 3 for exposed spores).

These two examples demonstrate the interest of AFM imaging for the fundamental elucidation of EP effect on cell morphology, shape, volume, and membrane ultrastructures.

Probing biophysical modifications induced by EP

More than an imaging tool, AFM is a force machine and when indenting the tip in the sample, nanomechanical parameters are measured. As already stated and observed on the AFM images, the mammalian cell cytoskeleton seems to be transiently destabilized by EP. The nanomechanical properties of the cells should therefore be altered too. To make this point sure, Chopinet et al. [START_REF] Chopinet | Destabilization induced by electropermeabilization analyzed by atomic force microscopy[END_REF] , in 2013 measured the Young Modulus of Chinese Hamster Ovary cells before EP and monitored the cell recovery across time. Figure 4A presents 5 elasticity maps of the same CHO cell across time. EP is applied at 0 time and the same area on the cell is monitored during 35 minutes after EP. Figure 4B shows indentation force curves where the applied force is plotted as a function of the tip indentation in the cell. It clearly demonstrates the softening of the cell after EP and the recovery 35 minutes after EP. Indeed, as seen on figure 4C (histogram in the center), the mean YM drops from 18.8 +/-2.0 kPa before EP to 11.2 +/-0.5 kPa just after EP. On fixed cells, the YM of native cells is 30.0 +/-2.0 kPa and decrease to 20.5 +/-1.0 kPa after EP. The same relative difference is thus found, emphasizing the pitfall that absolute values of YM can constitute, and the high attention that should be paid to the conditions of YM comparisons.

During the 35 minutes of the experiment on living cells, the authors show that the mean YM increases slowly to roughly reach the initial value again. The authors explored different areas of the cells according to their position in regard to the electrodes. The hypothesis was that areas facing the positive or the negative electrodes would be differently affected. However, no differences in the YM were found on these different areas of the cell membrane, suggesting a mechanism involving more than the cell membrane. It must be noticed that the mean YM is remarquably constant if no stress is applied to the cell (left histogram on figure 4 C).

Conversely the cell permeability induced by EP is shorter than the modification of the cells YM. The histogram on the right of figure 4 C demonstrates that 5 minutes after EP, the cell membrane is no more permeabilized and that small molecules like propidium iodide cannot anymore go through. (EP applied at time 0). unpaired t-test, *P value = 0.05; E: % of permeabilized cells across time after EP. adapted from [START_REF] Chopinet | Destabilization induced by electropermeabilization analyzed by atomic force microscopy[END_REF] Therefore, EP modifies the cell membrane but there is another mechanism responsible for the long-term modification of the cells YM. To explore the role of the actin cytoskeleton in this mechanism, the YM of latrunculin treated cells has been measured [START_REF] Louise | AFM sensing cortical actin cytoskeleton destabilization during plasma membrane electropermeabilization[END_REF] . Indeed, latrunculin is a drug inhibiting the actin polymerization. The YM of latrunculin treated cells was found to be close to the value of cells submitted to EP: 10.6 +/-0.7 kPa. Interstingly, removing the drug (latrunculin) from the medium results in an increase of the YM (figure 5D) toward values similar to cells having recovered from EP: 17.3 +/-1.7 kPa. The cytoskeleton fibers are reconstructed during this process as observed on both the AFM height images and the elasticity maps respectively on figure 5 A, B andE, F. Combining the two treatments (latrunculin and EP) provided surprising results. When applied first, latrunculin induces a dramatic decrease in the mean YM and EP, applied in a second step, has no more impact on the cell's elasticity. On the contrary, when applied first, EP induces the usual YM decrease and lactrunculin, applied in a second step has still an impact, reducing the mean YM of the cells toward 8.3 +/-0.3 kPa. This result means that EP and latrunculin both have an effect on the cytoskeleton, but not the same one! The author's interpretation is that EP destabilizes the cortical actin linked with the membrane whereas latrunclin inhibits the actin polymerization in the whole cell. This hypothesis was then reinforced by an experiment consisting in applying EP after removing latrunculin from the medium (figure 5I). In this case, cells were able to reconstruct their actin network but were unable to establish the link between the cortical actin network and the membrane, resulting in a low mean YM of 9.2 +/-0.6 kPa even 32 minutes after EP application. The AFM height image (figure 5C) and elasticity map (figure 5G) confirm that, even if the actin polymerization occurs, no cytoskeleton fibers can be observed. [START_REF] Louise | AFM sensing cortical actin cytoskeleton destabilization during plasma membrane electropermeabilization[END_REF] In 2014 Thompson et al. [START_REF] Thompson | Disruption of the actin cortex contributes to susceptibility of mammalian cells to nanosecond pulsed electric fields[END_REF] , used a similar strategy to explore the effect of nanosecond Pulsed Electric Fields (nsPEFs) (50 to 100 pulses of 10 ns; 150 kV/cm) on the actin cortex of cells.

In this work CHO-K1 cells elasticity is measured using a colloidal probe before and after EP, on native or latrunculin treated cells. Colloidal probes are AFM cantilevers terminated by a spherical bead (here made of borosilicate, and of 5 µm in diameter). The contact area between the probe and the sample is thus much higher than with pyramidal probes what is an advantage to measure global modifications, but is a drawback when it comes to measure fine structures like cytoskeleton fibers. Despite the different applied pulses and probe, they found quite similar results: nsPEFs induce a decrease in the cell elastic modulus of 50%. And Latrunculin A has the same qualitative effect but is quantitatively higher. The cell elasticity of latrunculin treated cells is, indeed, 75 % lower than for healthy cells. Interestingly, the application of nsPEFs on latrunculin treated cells had no effect on the cells elasticity. These results are in line with other reports already described here and confirm that EP has an action on the cells mechanical properties that is most probably mediated by modification at the cytoskeleton level.

In the case of microorganisms, the biophysical properties of the interface are driven by the cell wall 8 . To continue with Bacillus pumilus, which is a Gram-positive bacterium, the cell wall of the vegetative form is made of a thick (50 nm) layer of peptidoglycan. The peptidoglycan has been described to be coiled and super coiled around the cells 25 . The stiffness map presented in figure 6a adapted from [START_REF] Pillet | Cell wall as a target for bacteria inactivation by pulsed electric fields[END_REF] confirms this nanoscale organization of the peptidoglycan (doted lines correspond to peptidoglycan cables). After EP (figure 6d), the stiffness map does not show any peptidoglycan organization anymore. The mean stiffness did not really changed (increasing from 0.08 N/m before EP to 0.09 N/m after EP) but the surface became heterogeneous as evidenced by the standard deviation increasing from 0.01 N/m before EP to 0.03 after EP. In this work, the authors also studied the cell surface hydrophobicity as probed by chemical force microscopy (CFM) [START_REF] Dague | Chemical force microscopy of single live cells[END_REF]

Concluding remarks

Very few studies devoted to EP are using AFM. Works presented here nevertheless demonstrate the interest of this quite new technology in the field. Indeed, AFM has proven its ability to image the modifications induced by EP both on mammalian and bacteria cells.

Moreover it gives access to nanomechanical information, with a spatial resolution, that is crucial to better understand the fundamental mechanisms of EP. On bacteria the measure and mapping of adhesions, in this case of hydrophobicity, could have important applied consequences, eg in the food industry.

In the future it would be useful to develop a device to apply electric pulses on the AFM setup.

Then combining AFM and high resolution optical imaging it would be interesting to analyze the permeabilization of cells membranes toward anti-cancerous drugs or nucleic acids.

Figure 1 :

 1 Figure 1: Schematic representation of 3 AFM modes: contact, Force Volume (FV) and Quantitative imaging. These modes can be used to explore loosely immobilized sample, samples trap in holes of a Poly-Di-Methyl-Siloxane (PDMS) stamp or soft samples such as mammalian cells (adapted from 10 ).

Figure 2 :

 2 Figure 2: AFM QI images of living cells before and after EP. A 1-3: 3 AFM height images of the same untreated cell across time. B 1: AFM height image of an untreated cell; B 2-3: AFM height images of the cell presented in B1 respectively 18 and 50 minutes after EP. The scale bar in the insets represents 1µm. The insets present details of the membrane and are numerical zoom. C, D: cross sections of the cells taken along the lines indicated for the control cell (C) and the pulsed cell (D); the black line corresponds to non pulsed cells, light grey 15 minutes later and dark-grey 30 minutes later. The double arrow in D marks the cell swelling. The color scale is ranging from 0 to 7 µm. Adapted from 9

  presents AFM height images of vegetative cells before (a, c) and after EP (b, d).

Figure 3 :

 3 Figure 3: AFM QI images of Bacillus pumilus vegetative cells (a, b, c and d; scale bars: 200 nm) and spores (e, f, g and h; scale bars: 100 nm) before(a, b, e and f) and after (c, d, g and h) EP. The dotted scares indicate the position of imagesb, d, f, and h respectively on images a, c, e and g. The color scale, representing the highness, is given on the right of each image. Adapted from[START_REF] Pillet | Cell wall as a target for bacteria inactivation by pulsed electric fields[END_REF]

Figure 4 :

 4 Figure 4: Probing nanomechanical properties of electroporated cells across time. EP is applied at time 0, except on C where EP is not applied. A: YM maps of the same region acquired during 50 minutes (the color scale given on the right range fomr 0 to 36 kPa). B: typical force curves extracted from the maps presented in A. Black lines: before EP; light-grey line: 8 minutes after EP; dark-grey lines: 35 minutes after EP. C: evolution of the YM of a non puled cell across time; D: evolution of the YM of a pulsed cell

Figure 5 :

 5 Figure 5: Topography and elasticity of cells exposed to latrunculin and EP. A, B and C are AFM QI height images of a latrunculin treated cell (A), the same cell 35 minutes after latrunculin removal (B) and a cell 35 minutes after latrunculin removal but electroporated at time 0. E, F and G are the respective elasticity maps of the precedent conditions. The color scale given on the right of E range from 0 to 60 kPa. D: Evolution of the YM after latrunculin removal. Unpaired t test relative to the point before t=0: *P value<0.05; H: Evolution of the YM after luntrunculin removal if EP is applied at time 0. Adapted from 11

  (figure 6 b and h). They measured a mean adhesion force of 2.1 +/-1.2 nN (figure 6b) on untreated vegetative bacteria, 0.2 +/-0.3 nN on untreated spores (figure 6f), 0.2 +/-0.2 nN on pulsed vegetative cells (figure 6e) and 0.1 +/-0.1 nN on pulsed spores (figure 6f). The CFM technology clearly demonstrates in this study the impact of EP on both vegetative forms and spores of Bacillus pumilus hydrophobicity, which is a key parameter in adhesion and biofilm formation.

Figure 6 :

 6 Figure 6: AFM stiffness (a, d) and adhesion (b, c, e and f) maps recorded on Bacillus pumilus vegetative cells (a, b, d and e) and spores (e, f). The white bar represents 200 nm in a, b, d, e and 100 nm in c and f; the color scale next to each image corresponds to the stiffness (a, d) or the adhesion force (b, c, e and f) adapted from[START_REF] Pillet | Cell wall as a target for bacteria inactivation by pulsed electric fields[END_REF]