Rodolfo Orjuela
email: rodolfo.orjuela@ensem.inpl-nancy.fr

Benoît Marx
email: benoit.marx@ensem.inpl-nancy.fr

José Ragot
email: jose.ragot@ensem.inpl-nancy.fr

Didier Maquin
email: didier.maquin@ensem.inpl-nancy.fr

On the simultaneous state and unknown inputs estimation of complex systems via a multiple model strategy

Keywords: . nonlinear systems, multiple models, state estimation, unknown input multi-integral observer

This paper addresses the analysis and design of unknown input observer in order to provide both state and unknown input estimation of complex systems modelled with the help of a particular class of multiple model. The proposed observer uses the multi-integral strategy successfully employed in the classic linear control theory and known for its robustness properties. The observer design is based on the representation of the system via a multiple model, known as decoupled multiple model. This structure of multiple model allows to use submodels with different number of states and this fact constitutes the main advantage of the proposed observer with respect to the classic multiple model structure where the submodels have the same dimension. It is shown how the gains of the suggested observer can be obtained by solving a LMI optimal problem. An academic example is also proposed in order to illustrate the proposed methodology.

Introduction

In many real-world applications the direct measurement of the state variables of a system can be very difficult to obtain, or even impossible, due for example to physical constraints and/or economical restrictions. The use of an estimation of the state, using an observer, instead of its measured value, provided by sensors, is a solution largely adopted in order to avoid these problems. However, it is well known that in many situations some inputs of the system are inaccessible or considered for simplicity as such. These unmeasurable signals, denoted as unknown inputs (UI), have a serious impact on the state reconstruction and can be at the origin of biased estimations if they are not correctly taken into consideration.

Hence, both state and UI estimations have been intensively investigated for years because these estimations are of great use in several engineering applications such as control, supervision and fault-tolerant control. Indeed, an actuator/sensor failure or an abnormal behaviour of an internal component of the system can be regarded as unmeasured signals modelled by UIs. Hence, the UI estimations can be considered as fault symptoms of the system for fault detection and isolation in order to improve the safety of the system.

The so-called unknown input observer (UIO) is usually employed as a mean to provide both state and UI estimations of a system. There are several approaches for unknown input observer design for system represented by linear time-invariant models. In a general way, the UIO design consists in finding a projection operator or a matrix transformation fulfilling a set of algebraic equations in order to decouple the UI and the estimation error (see [START_REF] Hou | Design of observers for linear systems with unknown inputs[END_REF][START_REF] Gaddouna | Fault detection observer for systems with unknown inputs[END_REF][START_REF] Darouach | Full-order observers for linear systems with unknown inputs[END_REF][START_REF] Corless | State and input estimation for a class of uncertain systems[END_REF][START_REF] Hui | Observer design for systems with unknown inputs[END_REF] and references therein). Note however that, as pointed out in [START_REF] Hou | Input observability and input reconstruction[END_REF], output derivatives are required in order to perform an UI estimation. Undoubtedly the sensor noises are unavoidable and derivative of such signals may be unsuitable.

Another attractive way to obtain a simultaneous estimation of the states and the UI is to use the Proportional-Integral observer (PIO). This observer known for its robustness properties [START_REF] Beale | Robust control system design with a proportional integral observer[END_REF][START_REF] Weinmann | Uncertain Models and Robust Control[END_REF] has been successfully employed in order to cope with the state and the UI estimations when the UI is characterised by low frequency signals (constant or slowly varying signals) [START_REF] Söffker | State estimation of dynamical systems with nonlinearities by using proportional-integral observer[END_REF][START_REF] Busawon | Disturbance attenuation using proportional integral observers[END_REF][START_REF] Shafai | Simultaneous disturbance attenuation and fault detection using proportional integral observers[END_REF][START_REF] Xiong | Unknown disturbance inputs estimation based on a state functional observer design[END_REF][START_REF] Marx | Robust fault diagnosis for linear descriptor systems using Proportional Integral observers[END_REF]. Recently, in [START_REF] Jiang | Design of observer with integrators for linear systems with unknown input disturbances[END_REF] the concept of PIO has been generalised to Multiple-Integral Observer (MIO), by replacing the integral action by a chain of integral actions. Thanks to these extra integral actions, the MIO is able to provide robust state estimations when the considered UI takes a polynomial form which is more general than the considered constant unknown input. Recently, other implementation schemes based on the MIO principle and their extension to Lipschitz nonlinear systems with single output are investigated in [START_REF] Ibrir | Robust state estimation with q-integral observers[END_REF]. For generalised linear time-invariant state space systems (or descriptor system), the MIO design is addressed in [START_REF] Gao | Proportional multiple-integral observer design for descriptor systems with measurement output disturbances[END_REF][START_REF] Koenig | Unknown input proportional multiple-integral observer design for linear descriptor systems: application to state and fault estimation[END_REF].

Moreover, nowadays the complexity of many physical systems is undoubtedly increased involving nonlinear dynamic behaviours which are governed by complex physical laws. Hence, the use of a single linear model for modelling the dynamic behaviour of such systems in the whole operating space is unsuited because a linear model provides only a local approximation in a small neighbourhood of an operating point (local modelling). On the other hand, rigorous modelling of complex systems in a wide operational range (global modelling) can be in practice very difficult and even if this modelling is possible the available model takes typically a nonlinear form which cannot be generally used in a systematic way for designing an observer. However, a global representation of such systems can effectively be expressed by blending judiciously a set of linear local models which describes the dynamics of the system in some small region of the operating space. In this modelling framework, the obtained model is known as local model network or multiple model (see [START_REF] Murray-Smith | Multiple model approaches to modelling and control[END_REF][START_REF] Leith | Analytic framework for blended multiple model systems using linear local models[END_REF][START_REF] Gao | Multiple model networks in non-linear systems model for control -a review[END_REF] and references therein). From a practical point of view, a multiple model is built by reducing the system complexity via a decomposition of its operating space in a finite number of operating zones within which the dynamics are characterised by a local model, also called submodel, with relatively simple structure, often linear. The global dynamic behaviour of the system is finally captured by considering the relative contribution of each submodel by means of a weighting function associated to each operating zone. In brief, multiple model approach provides a suitable framework for modelling nonlinear systems by an association of a set of submodels blended throughout an interpolation scheme.

It should be noted that the interpolation scheme used for blending the submodels plays an important role in this context. Indeed, the same set of submodels can be associated in various ways which provide several kinds of blended multiple models. However as pointed out in [START_REF] Filev | Fuzzy modeling of complex systems[END_REF], two main structures of multiple models can be distinguished. In the first one, the multiple model is made up of homogeneous submodels in the sense that the submodels have the same structure and share the same state space across the operating space. In the second one, heterogeneous submodels can be used because their state spaces are decoupled. It should be noted that these two multiple model realisations don't have an identical dynamic behaviour.

The so-called Takagi-Sugeno multiple model is the main example of homogeneous multiple models. It has been initially proposed, in a fuzzy modelling framework, by Takagi and Sugeno [START_REF] Takagi | Fuzzy identification of systems and its applications to model and control[END_REF] and in a operating regimebased modelling framework by Johansen and Foss [START_REF] Johansen | Constructing NARMAX models using ARMAX model[END_REF]. This multiple model has been extensively considered in the literature for modelling, control and state estimation of nonlinear systems [START_REF] Murray-Smith | Multiple model approaches to modelling and control[END_REF][START_REF] Babuska | Fuzzy modeling for control[END_REF][START_REF] Tanaka | Fuzzy Control Systems design and analysis[END_REF]. In this multiple model representation, the parameters of the submodels are combined by means of the weighting functions and then a common state vector appears in the dynamic equation of the overall model. It should be noted that this same multiple model structure is encountered in a number of quite diverse similar modelling frameworks such as piecewise linear model, polytopic models, hybrid systems, switched systems, markovian switching systems, etc.

Despite different names, these approaches share the same modelling philosophy. In these different frameworks, the problem of estimating the state and the UI is tackled in several ways. For systems represented by Takagi-Sugeno multiple models, Luenberger observers including a sliding term to compensate the effect of the unknown inputs are used in [START_REF] Akhenak | Estimation of state and unknown inputs of a non linear system represented by a multiple model[END_REF]. For polytopic models, a polytopic unknown input observer design for providing actuator fault estimation in active fault tolerant control strategy is proposed in [START_REF] Rodrigues | Design of an active fault tolerant control and polytopic unknown input observer for nonlinear systems described by a multi-model representation[END_REF]. For markovian switching systems a finite memory observer is considered in order to cope with this estimation problem [START_REF] Hocine | Finite memory observer for switching systems: Application to diagnosis[END_REF]. For hybrid systems an extension of the moving horizon estimation using a transformation of the original PieceWise Affine (PWA) system into Mixed Logical and Dynamical (MLD) systems is proposed in [START_REF] Pina | Simultaneous state and input estimation of hybrid systems with unknown inputs[END_REF]. Nevertheless, these contributions are proposed on the basis of homogeneous local models across the operating space.

The second multiple model realisation , known as decoupled multiple model, can be used in order to take into consideration heterogeneous submodels. This model introduce some degree of flexibility in the modelling stage, since the dimension (e.g. the number of state variables) of each submodel can be adapted to the complexity of the system inside each operating zone and this constitutes the originality of the proposed approach (details are given in section 2). Nonlinear systems identification using decoupled multiple model is addressed in [START_REF] Venkat | Identification of complex nonlinear processes based on fuzzy decomposition of the steady state space[END_REF][START_REF] Vinsonneau | Extended global total least square approach to multiple-model identification[END_REF][START_REF] Orjuela | Nonlinear system identification using uncoupled state multiple-model approach[END_REF] and the control laws design in [START_REF] Gawthrop | Continuous-time local state local model networks[END_REF][START_REF] Gatzke | Multiple model approach for CSTR control[END_REF][START_REF] Gregorcic | Control of highly nonlinear processes using self-tuning control and multiple/local model approaches[END_REF][START_REF] García-Nieto | Nonlinear predictive control based on local model networks for air management in diesel engines[END_REF]. It should be noted that the state estimation problem of a nonlinear system using this multiple model has been poorly considered in the literature [START_REF] Uppal | A neuro-fuzzy multiple-model observer approach to robust fault diagnosis based on the DAMADICS benchmark problem[END_REF][START_REF] Orjuela | State estimation for nonlinear systems using decoupled multiple model[END_REF]. In [START_REF] Uppal | A neuro-fuzzy multiple-model observer approach to robust fault diagnosis based on the DAMADICS benchmark problem[END_REF] the state estimation problem is addressed in order to setting up a fault diagnosis strategy. However just a note on the estimation error convergence is proposed by the authors and no analytic proof of the convergence of the estimation error is given. These works show the relevance and the successful implementation of this structure for state estimation.

The main contribution of this paper is an extension of MIO design for linear systems to nonlinear discrete-time systems represented by decoupled multiple models. The MIO is designed in order to provide both state and UI estimations by minimising the influence of external disturbances on the estimation error. An analytic proof of the convergence of the estimation errors is clearly established by using the well known Lyapunov theory. The robust L 2 existence conditions of the MIO are expressed in the form of a set of linear matrix inequalities (LMIs) [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF].

So far, the proportional and integral observer (PIO) design for linear discrete time systems seems only reported in some recent papers [START_REF] Chang | Applying discrete-time proportional integral observers for state and disturbance estimation[END_REF][START_REF] Gao | Discrete-time proportional-integral observer and observer based controller for systems with unknown disturbances[END_REF]. Hence, the proposed results can also be used for designing a MIO in the single model linear case because single model and PIO are a particular case of multiple models and MIO.

The outline of this paper is as follows. The decoupled multiple model representation is presented in section 2.

Preliminaries and the suggested MIO are presented in section 3. In section 4, robust L 2 observer design is proposed and the gains of the observer are obtained by LMI optimization. The last section gives a simulation example to illustrate the effectiveness of the proposed approach.

Notations. The following standard notations will be used. P > 0 (P < 0) denotes a positive (negative) definite matrix P; X T denotes the transpose of matrix X, I is the identity matrix of appropriate dimension and diag{A 1 , ..., A n } stands for a block-diagonal matrix with the matrices A i on the main diagonal. The L 2 -norm of a signal, quantifying its energy, is denoted and defined by e(k) 2 2 = ∞ ∑ 0 e T (k)e(k). Finally, we shall simply write µ i (ξ (k)) = µ i (k).

On the decoupled multiple model representation

Multiple model framework is an attractive way in the field of complex systems modelling because a large class of nonlinear dynamic behaviours can be captured using this representation. Note also that multiple model makes it possible the partial extension of some results obtained in the linear control theory to nonlinear systems avoiding specific analysis of the non-linearity of the system. In brief, multiple model offers good accuracy representation by means of an usable model.

The structure of the decoupled multiple model, firstly proposed in [START_REF] Filev | Fuzzy modeling of complex systems[END_REF], is here slightly modified using a state space representation as follows:

x i (k + 1) = A i x i (k) + B i u(k) + D i η(k) +V i w(k) , (1a)
y i (k) = C i x i (k) + E i η(k) , (1b)
y(k) = L ∑ i=1 µ i (ξ (k))y i (k) +W w(k) , (1c)
where x i ∈ R n i and y i ∈ R p are respectively the state vector and the output of the i th submodel; u ∈ R m is the known input vector, η ∈ R l the UI vector, y ∈ R p the measured output and w ∈ R r a disturbance (noise, etc.). The matrices

A i ∈ R n i ×n i , B i ∈ R n i ×m , D i ∈ R n i ×l , V i ∈ R n i ×r , C i ∈ R p×n i , E i ∈ R p×l and W ∈ R
i = [ Di 0], E i = [0 E] and η(k) = [η a (k) η s (k)] T .
Note that a common matrix E i is used for modelling an UI acting on the multiple model output (e.g. a sensor fault).

The so-called decision variable ξ (k) is used in order to take into account the current operating point of the system. It is assumed known and real-time accessible, currently the inputs and/or measured variables of the system are employed as decision variable. The relative contribution of each submodel according to the operating point of the system is quantified by the weighting functions µ i (ξ (k)) which satisfy the following convex sum constraints:

L ∑ i=1 µ i (ξ (k)) = 1 and 0 ≤ µ i (ξ (k)) ≤ 1 , ∀i = 1...L, ∀k. (2)
The role of the weighting functions is to allow a transition, often smooth, between the contribution of the submodels. Hence the contribution of several submodels can be taken into account at the same time because the weighting functions take intermediary values over the range 0 to 1. So the dynamic behaviour of the multiple model can therefore be considered as truly nonlinear instead of a piecewise linear behaviour.

As it can clearly be seen from equation ( 1), the submodels are run using a parallel scheme and the multiple model output is obtained via a weighted sum of the submodel outputs. Therefore, the submodels do not share the same state space and consequently their dimension (i.e. the number of state variables) and structure can be different across the operating space of the system. Hence, it can be expected that the decoupled multiple model accurately describes nonlinear systems with a relatively small number of parameters. Indeed, the use of heterogeneous submodels provides flexibility in the modelling stage because each submodel can be well adapted to the complexity of the system inside each operating zone. In a black box modelling, this feature can be used in order to cope with the so-called curse of dimensionality problem where the number of parameters needed for an accurate representation increases extremely rapidly as the order of the nonlinear dynamic system increases. This multiple model structure is then suited for a black box modelling of complex systems with variable structure in the operating range. Note that the local outputs y i (k) of the submodels are "artificial modelling signals" only used to provide a representation of the real system behaviour but can neither be related physically to the true system nor measured.

Preliminaries

Firstly, the aim of this section is to introduce a compact rewriting of the decoupled multiple model in order to reduce the further mathematical manipulation. Secondly, based on this new compact form, the proposed MIO is presented and the estimation errors are established. Finally, a brief discussion about sufficient conditions for the UI decoupled state estimation is proposed.

Compact representation of the multiple model

Notice that by using the following augmented state vector:

x(k) = x T 1 (k) • • • x T i (k) • • • x T L (k) T ∈ R n , n = L ∑ i=1 n i , (3)
the decoupled multiple model ( 1) can be rewritten in the following compact form:

x(k + 1) = Ãx(k) + Bu(k) + Dη(k) + Ṽ w(k) , (4a)
y(k) = C(k)x(k) + Ẽ(k)η(k) +W w(k) , (4b)
where

à = diag {A 1 • • • A i • • • A L } , (5)
B = B 1 T • • • B i T • • • B L T T , (6)
D = D 1 T • • • D i T • • • D L T T , (7)
Ṽ = V 1 T • • • V i T • • • V L T T , (8)
C(k) = µ 1 (k)C 1 • • • µ i (k)C i • • • µ L (k)C L , (9)
Ẽ(k) = L ∑ i=1 µ i (k)E i . (10)
The reader may have noticed that the time-varying matrix C(k) can be rewritten as follows:

C(k) = L ∑ i=1 µ i (k) Ci , (11)
where Ci is a constant block matrix given by:

Ci = 0 • • • C i • • • 0 (12)
such as the term C i is found on the i th block column of Ci .

Multi-Integral Observer presentation

Assuming that the UI acting on the system is modelled by a polynomial form of degree f in the variable k as follows

η(k) = Q 0 + Q 1 k + Q 2 k 2 + • • • + Q f k f . ( 13
)
From a practical point of view, a wide class of UIs can be modelled via a polynomial function (constant signal, ramps, etc.). Let us notice that the polynomial degree f of the unknown input is the only information supposed available, the coefficients Q f of the polynomial are assumed unknown.

The following state observer on the basis of the model ( 4) and the UI ( 13) is proposed with the aim of providing a simultaneous estimation of the state and the UI:

x(k + 1) = Ã x(k) + Bu(k) + D η0 (k) + K p (y(k) -ŷ(k)) , (14a) ŷ
(k) = C(k) x(k) + Ẽ(k) η0 (k) (14b)
coupled to the following UIO:

ηi (k + 1) = ηi (k) + K i (y(k) -ŷ(k)) + ηi+1 (k), i = 0, ..., q -1 , (15a) ηq
(k + 1) = ηq (k) + K q (y(k) -ŷ(k)) , (15b)
where x(k) and η0 (k) are the estimates of x(k) and η(k) respectively.

As can be shown in the figure 1, the UI estimation is obtained using the multi-integral strategy (in this figure the operator z -1 is the one step delay operator). Indeed, the estimation of the UI is performed using a recurrent schema given by a chain of integral actions. Hence, the inputs of the i -1 th block are the output estimation error given by y(k)ŷ(k) and the output of the i th block.

Definitions of the estimation errors

In this section, the state and UI estimation errors are studied to prove that x(k) and η0 (k) converge toward x(k) and η(k) respectively. In order to establish the estimation errors the difference operator must be introduced.

Definition 1 (Difference operator).

The first difference of a function ϕ(k) is a function defined by:

∆ϕ(k) ϕ(k + 1) -ϕ(k) . ( 16
)
The q th -difference operator is given by:

∆ (q) ϕ = ∆(∆ (q-1) ϕ(k)) . ( 17
)
For example, the second difference of a function ϕ(k) is given by:

∆ (2) ϕ = ∆(ϕ(k + 1) -ϕ(k)) , ∆ (2) ϕ = ϕ(k + 2) -2ϕ(k -1) + ϕ(k) .
Now, the state and UI estimation errors are defined by:

e(k) = x(k) -x(k) , (18)
ε i (k) = ∆ (i) η(k) -ηi (k), i = 0, ..., q , (19)
where

∆ (0) η(k) = η(k).
Remark 3. The architecture of the proposed observer allows us to obtain an estimation of the state, the unknown input and its successive q th differences (i.e. the coefficients of the polynomial) at the same time.

Dynamic equations of these errors must be now established. Hence, dynamics of the state estimation error is given by:

e(k + 1) = ( Ã -K p C(k))e(k) + ( D -K p Ẽ(k))ε 0 (k) + ( Ṽ -K p W )w(k) . (20)
The above equation is easily obtained with the help of equations ( 4), ( 14), ( 18) and ( 19) by considering i = 0. Let us notice that the state estimation error is directly affected by the UI estimation error ε 0 (k) and the disturbance w(k) acting on the system.

The dynamics of the UI estimation error, ε i (k) for 0 ≤ i < q , is obtained using ( 19) and (15):

ε i (k + 1) = ε i (k) + ε i+1 (k) -K i C(k)e(k) -K i Ẽ(k)ε 0 (k) -K i W w(k), i = 0, ..., q -1 , (21)
and finally the dynamics of ε q (k) is given by:

ε q (k + 1) = ε q (k) -K q C(k)e(k) -K q Ẽ(k)ε 0 (k) -K q W w(k) + ∆ (q+1) η(k) . ( 22
)
We can rewrite in a compact form the dynamics of the state estimation error, the UI estimation error and their successive differences using an augmented error vector given by:

Σ(k) = e T (k) ε T 0 (k) . . . ε T i (k) . . . ε T q (k) T , (23)
as follows:

Σ(k + 1) = (Λ -KΓ(k))Σ(k) + (V -KW )w(k) + Φ∆ (q+1) η(k) , (24)
where

Λ =                 Ã D
                , K =                 K p K 0 . . . K i . . . K q                 , Φ =                 0 0 0 . . . . . . I                 , V =                 Ṽ 0 0 . . . . . . 0                 , Γ(k) =                 C(k) T Ẽ(k) T 0 . . . . . . 0                 T . (25)
Notice that by using the definition [START_REF] Shafai | Simultaneous disturbance attenuation and fault detection using proportional integral observers[END_REF] of C(k) and the definition (10) of Ẽ(k), the time-varying matrix Γ(k) can be rewritten as:

Γ(k) = L ∑ i=1 µ i (k)Ω i . ( 26
)
where

Ω i = Ci Ẽi 0 . . . . . . 0 . (27)
Finally, taking into consideration the previous matrix transformation, the equation ( 24) becomes:

Σ(k + 1) = Ψ(k)Σ(k) + (V -KW )w(k) + Φ∆ (q+1) η(k) , (28)
Ψ(k) = L ∑ i=1 µ i (k)(Λ -KΩ i ) . (29)
Conceptually, the aim of the design is to determine an augmented gain matrix K ∈ R n+(q+1)l×p that guarantees the asymptotic convergence of the estimation error towards zero (detailed problem formulations are presented in the section 4).

Discussion about unknown input decoupling

It can be seen, from equation [START_REF] Hocine | Finite memory observer for switching systems: Application to diagnosis[END_REF], that the estimation error is totally decoupled from the unknown input (i.e.

the influence of the unknown input on the estimation error Σ(k) is vanished) if and only if ∆ (q+1) η(k) = 0. This conditions can be accomplished when the two following conditions are simultaneously satisfied:

1. the unknown input η(k) takes a polynomial form of degree f in the variable k, 2. the number of integral actions q taken into account is at least equal to f + 1.

Hence the number of integral actions q in the observer depends on the polynomial degree f taken into account for modelling the UI. Note that the minimal number of integral actions needed for ensuring the unknown input decoupling is fixed by the second condition. For example, if the UI is modelled by polynomial of degree zero η(k) = Q 0 then only one integral action is necessary for ensuring ∆η(k) = 0. If a polynomial of degree one is considered η(k) = Q 0 + Q 1 k then two integral actions are necessary for ensuring ∆ 2 η(k) = 0, and so on.

Problem formulations and main results

Three main problems can be examined in the purpose to simultaneously estimate the state and the unknown input using the proposed MIO:

1. Problem 1. Obtain conditions for ensuring both state and UI estimation error convergence, i.e. the observer stability, when ∆ (q+1) η(k) = 0 and w(k) = 0.

2. Problem 2. Obtain conditions for both state and UI estimation error convergence by considering a perturbed multiple model i.e. w(k) = 0. This problem can be viewed as the robust observer design with respect to disturbances.

3. Problem 3. Obtain conditions for both state and UI estimation error convergence when the UI is not truly a polynomial form i.e. ∆ (q+1) η(k) = 0 and by considering w(k) = 0.

Let us notice that in the multiple model framework, an unstable multiple model can be obtained by blending a set of stable submodels. Hence the stability assumption of the submodels does not guarantee the stability of the multiple model. Consequently, the individual stability of matrices Λ -KΩ i is not sufficient in order to guarantee the stability of the time-varying ψ(k) given by [START_REF] Pina | Simultaneous state and input estimation of hybrid systems with unknown inputs[END_REF]. Hence, a classic observer design for each submodel cannot be performed independently: the blending between the submodels must be taken into consideration in the observer design in order to ensure the convergence of the estimation error [START_REF] Johansen | Constructing NARMAX models using ARMAX model[END_REF]. Therefore, the analysis of these problems will be carried out in the time-domain with the help of the Lyapunov method and the proposed solutions for the observer design are derived in terms of LMIs [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF].

Problem 1: convergence conditions of the estimation error

Convergence conditions of the estimation error, in the disturbance free case and with an UI perfectly decoupled, will be established under the following assumption:

Assumption 1. w(k) = 0 and ∆ (q+1) η(k) = 0.
The following theorem presents the convergence conditions:

Theorem 1. Consider the model (4) and assumption 1. There exists an observer [START_REF] Jiang | Design of observer with integrators for linear systems with unknown input disturbances[END_REF] and [START_REF] Ibrir | Robust state estimation with q-integral observers[END_REF] such that the state and the UI estimation error [START_REF] Johansen | Constructing NARMAX models using ARMAX model[END_REF] asymptotically converges towards zero if there exist P = P T > 0 and M such that:

   P PΛ -MΩ i (PΛ) T -(MΩ i ) T P    > 0, i = 1...L . ( 30
)
The observer gains are given by K = P -1 M where K is defined by [START_REF] Tanaka | Fuzzy Control Systems design and analysis[END_REF].

Conceptually, the conditions of this theorem ensure the stability of the system defined in [START_REF] Hocine | Finite memory observer for switching systems: Application to diagnosis[END_REF] for any blend between the submodel outputs and for any initial conditions.

Proof. Define the time dependent quadratic Lyapunov function by:

V (k) = Σ T (k)PΣ(k), (31)
where P = P T > 0. The variation of the above function is given by:

∆V (k) = V (k + 1) -V (k), (32)
which must be negative in order to ensure the asymptotic convergence towards zero of the estimation error.

By considering the assumption 1 and ( 28) then (32) becomes:

∆V (k) = Σ T (k){Ψ T (k)PΨ(k) -P}Σ(k) , (33)
that is a quadratic form in Σ(k). The negativity of ( 33) is then ensured if:

Ψ T (k)PΨ(k) -P < 0 , (34)
which can be rewritten, using the Schur complement and (29), as follows:

L ∑ i=1 µ i (k)    P P(Λ -KΩ i ) P(Λ -KΩ i ) P    > 0 . (35)
Now, according to the convex sum properties (2), the above inequality is also verified if the following inequalities hold:

   P P(Λ -KΩ i ) (Λ -KΩ i ) T P P    > 0 , i = 1...L . ( 36
)
Let us notice that the above inequalities do not take a LMI form in the variables P and K, therefore the classical LMI tools cannot be directly used. However, they become a LMI by setting M = PK and the proof is completed.

Problem 2: convergence conditions in the presence of disturbances

Now, model ( 4) is assumed to be affected by energy-bounded disturbances but the UI remains perfectly decoupled.

In other words the following assumptions are made:

Assumption 2. The disturbance w(k) is such that w(k) 2 2 < ∞. Assumption 3. The UI is such that ∆ (q+1) η(k) = 0.
The robust observer design problem can thus be formulated as finding a matrix gain K such that the objective signal z(k) to be attenuated and defined by:

z(k) = HΣ(k) , (37)
satisfies the following design objectives:

lim k→∞ Σ(k) = 0 for w(k) = 0 , (38a) z(k) 2 2 ≤ γ 2 w(k) 2 2 for w(k) = 0 and z(0) = 0 , ( 38b
)
where γ is the L 2 gain from w(k) to z(k). Notice that the convergence of the estimation error in the disturbance free case is ensured by (38a) and robust state estimation in presence of a disturbance is ensured by (38b). Finally, note that thanks to matrix H in [START_REF] Uppal | A neuro-fuzzy multiple-model observer approach to robust fault diagnosis based on the DAMADICS benchmark problem[END_REF], the attenuation level is guaranteed by considering partially or totally the components of the augmented error Σ(k).

The following theorem presents robust convergence conditions:

Theorem 2 (Robust convergence conditions). Considerer the model (4) and assumptions 2 and 3. There exists an observer [START_REF] Jiang | Design of observer with integrators for linear systems with unknown input disturbances[END_REF] and [START_REF] Ibrir | Robust state estimation with q-integral observers[END_REF] ensuring the objective [START_REF] Orjuela | State estimation for nonlinear systems using decoupled multiple model[END_REF] if there exist P = P T > 0, M and a scalar γ > 0 such that:

      -P PΛ -MΩ i PV -MW (PΛ -MΩ i ) T -P + H T H 0 (PV -MW ) T 0 -γ 2 I       < 0, i = 1...L , ( 39
)
for a prescribed matrix H. The matrix gain is given by K = P -1 M.

Remark 4.

The attenuation level γ can be minimized by considering γ = γ 2 as an LMI variable to be minimized under the constraints γ > 0, P = P T > 0 and [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF].

Proof. In order to ensure robust performances defined by [START_REF] Orjuela | State estimation for nonlinear systems using decoupled multiple model[END_REF], let us consider the quadratic Lyapunov function [START_REF] Vinsonneau | Extended global total least square approach to multiple-model identification[END_REF], its variation ∆V (k) = V (k + 1) -V (k) and γ > 0 such that [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]:

∆V (k) < -z T (k)z(k) + γ 2 w T (k)w(k) , ∀k . (40)
It is easily established that, by considering V (0) = 0, (40

) implies z(k) 2 2 < γ 2 w(k) 2 2 .
Hence, robust performances [START_REF] Orjuela | State estimation for nonlinear systems using decoupled multiple model[END_REF] are achieved by satisfying [START_REF] Chang | Applying discrete-time proportional integral observers for state and disturbance estimation[END_REF]. To that purpose, consider (28) and assumptions 2 and 3, the variation of the Lyapunov function is then given by:

∆V (k) = Σ T (k)Ψ T (k)PΨ(k)Σ(k) + Σ T (k)Ψ T (k)P(V -KW )w(k) + w T (k)(V -KW ) T PΨ(k)Σ(k) + w T (k)(V -KW ) T P(V -KW )w(k) -Σ T (k)PΣ(k) , (41)
which can be rewritten as:

∆V (k) = Φ T (k)Ω(k)Φ(k) , (42)
where:

Ω(k) =    Ψ T (k)PΨ(k) -P Ψ T (k)P(V -KW ) (V -KW ) T PΨ(k) (V -KW ) T P(V -KW )    , (43)
Φ(k) = Σ T (k) w T (k) T . (44)
Finally, ( 40) is ensured if

Φ T (k)      Ω(k) +    H T H 0 0 -γ 2 I         Φ(k) < 0 , (45)
which is a quadratic form in Φ(k). This inequality holds provided the expression inside the brackets is negative definite:

   Ψ T (k)PΨ(k) -P + H T H Ψ T (k)P(V -KW ) (V -KW ) T PΨ(k) (V -KW ) T P(V -KW ) -γ 2 I    < 0 , (46)
which becomes using the Schur complement:

      -P PΨ(k) P(V -KW ) Ψ T (k)P -P + H T H 0 (V -KW ) T P 0 -γ 2 I       < 0 . ( 47
)
The proof is completed as in the previous case by using [START_REF] Pina | Simultaneous state and input estimation of hybrid systems with unknown inputs[END_REF], the convex sum properties of µ i (k) given by ( 2) and the change of variables M = PK.

Problem 3: state and UI estimation when the (q + 1) th -difference of the UI is not null

In this section, a perturbed decoupled multiple model without a total UI decoupling is considered (assumption 3 is not verified). Indeed, the (q + 1) th -difference of the unknown input is not assumed to be null but bounded by an unknown function ∆ (q+1) η(k) ≤ δ (k). Note that the solution of this problem can be useful in order to reduce the number of integral actions.

Assumption 4. The UI satisfies ∆ (q+1) η(k) = δ (k) with δ (k) 2 2 < ∞.
The estimation error equation ( 28) can be re-written as:

Σ(k + 1) = Ψ(k)Σ(k) + Πθ (k) , (48)
where

Π = V -KW Φ and θ (k) = w T (k) δ T (k) T . (49)
Thus the unknown function δ (k) is considered as a disturbance. Now, this problem can be regarded in a similar way that the previous one. Hence the theorem 2 can be directly used by considering θ (k) as disturbance instead of w(k).

The robust observer design problem can thus be formulated as finding a matrix gain K such that the objective signal z(k) to be attenuated and defined by:

z(k) = HΣ(k) , (50)
satisfies the following design objectives:

lim k→∞ Σ(k) = 0 for w(k) = 0 , (51a)
z(k) 2 2 ≤ γ 2 θ (k) 2 2,Q for θ (k) = 0 and z(0) = 0 , (51b)
where γ is the L 2 gain from θ (k) to z(k). Note that here a weighted

L 2 -norm defined by θ (k) 2 2,Q = ∞ ∑ 0 θ T (k)Qθ (k)
where Q > 0 is employed in order to weight the relative importance given to w and δ .

The following theorem presents robust convergence conditions:

Theorem 3 (Robust convergence conditions). Considerer the model (4) and assumptions 2 and 4. There exists an observer [START_REF] Jiang | Design of observer with integrators for linear systems with unknown input disturbances[END_REF] and [START_REF] Ibrir | Robust state estimation with q-integral observers[END_REF] ensuring the objective (51), if there exist P = P T > 0, M and scalar γ > 0 such that:

      -P PΛ -MΩ i Π (PΛ -MΩ i ) T -P + H T H 0 ΠT 0 -γ 2 Q       < 0, i = 1...L , where Π = PV -MW PΦ ,
for prescribed matrices H and Q > 0. The matrix gain is given by K = P -1 M.

Proof. The proof of this theorem is immediate from theorem 2.

Simulation example

In this section two examples are proposed in order to illustrate simultaneous state and UI estimation in presence of disturbances (problem 2) and in the case when the (q + 1) th -difference of the UI is not assumed to be null (problem 3).

Consider the model ( 4), made up of two different dimension submodels and involving two measured outputs and two unknown inputs. Here, the decision variable ξ (k) is the input signal u(k) ∈ [0, 1] with sample time equal to T s = 0.01. The weighting functions are obtained from normalised gaussian functions:

µ i (ξ (k)) = ω i (ξ (k))/ L ∑ j=1 ω j (ξ (k)) , (52)
ω i (ξ (k)) = exp -(ξ (k) -c i ) 2 /σ 2 , , (53)
with σ = 0.4 and the centres c 1 = 0.25 and c 2 = 0.75.

The numerical values of the matrices of the submodels are as follows:

A 1 =    -0.5 -0.7 0.4 0.1    , A 2 =       -0.7 0.2 0.5 0.3 -0.4 -0.1 -0.2 -0.3 0.6       , B 1 =    1 -0.8    , B 2 =       0.2 0.3 0.4       , C 1 =    0.7 0.4 -0.5 0.3    , C 2 =    0.3 0.0 0.5 -0.6 -0.2 -0.4    , D 1 =    0.2 0.0 0.1 0.0    , D 2 =       0.1 0.0 0.2 0.0 0.3 0.0       , E 1 = E 2 =    0.0 0.4 0.0 0.5    , V 1 =    0.2 0.3 0.1 -0.2    , V 2 =       0.2 0.3 0.1 -0.2 -0.1 0.2       , W =    -0.2 0.1 0.1 -0.2    .
The disturbance w(k) and the UI η(k) take respectively the following forms:

w(k) =    0.04 sin(40k) 0.035 sin(60k)    and η(k) =    η a (k) η s (k)    . (54)
In this example the UI are faults acting on the system. Note the particular form of D i , E i and η(k) which allows us to take into consideration UIs affecting independently the dynamics (e.g. an internal component failure) and the overall output of the multiple model (e.g. a sensor fault).

On the other hand, note that the eigenvalues of the submodels are given by : λ 1 = -0.2 ± 0.43i and λ 2 = -0.83 -0.16 0.50 , thus, the dynamics of the submodels are different and the dynamic behaviour of the multiple model can be expected nonlinear. The input, outputs and the weighting functions of the multiple model in the UI-free case are plotted in figure 2. One should note that, at every time, both submodels are taken into consideration to compute the global model.

State and UI estimation in presence of disturbances: problem 2

In this section, faults η a (k) and η s (k) are defined by:

η a (k) =                0 if 0 ≤ kT s ≤ 5, 0.1333k -0.6665 if 5 ≤ kT s ≤ 6.5, 0.2 otherwise, and η s (k) =                0 if 0 ≤ kT s ≤ 5, -0.35 if 5 ≤ kT s ≤ 8, -0.15 otherwise. (55)
A total decoupling can be accomplished by using a MIO composed of two integral actions because η a is piecewise polynomial of maximum degree equal to one and η s is a piecewise constant signal.

The observer design is based on LMI conditions of theorem 2 by considering the attenuation level γ as unknown variable to be minimised (see remark 4). In this example, the objective signal z(k) is the state estimation error of the submodels then H = [I (5×5) 0 (5×4) ]. Hence, only the L 2 -gain between the disturbances and the state estimation error will be attenuated in the observer design procedure.

The optimal solution satisfying conditions of theorem 2 is found by using the YALMIP interface [START_REF] Löfberg | YALMIP : A toolbox for modeling and optimization in MATLAB[END_REF] coupled to SeDuMi solver [START_REF] Sturm | Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones[END_REF]. Thus, conditions of theorem 2 are fulfilled with:

K =    -0.
   T , (56)
with a minimal attenuation level given by γ = 3.1623.

State variables of the multiple model and state estimation errors provided by the proposed observer are plotted in figures 3 and 4. In the simulation, the initial conditions of the multiple model are not null and the initial conditions of the observers are null. The estimation error provided by the MIO is not globally affected by the UIs and remains globally bounded despite the disturbances and UIs acting on the system. Let us notice that the amplitudes of the obtained estimation errors are admissible with respect to the amplitudes of the state variables. Besides, the state estimation is punctually affected when the value of the UIs changes abruptly, for example at kT s = 5 and kT s = 8, obviously in this abrupt transition the UI is not a polynomial. However, the estimation errors provided by the proposed MIO have a good transient response and good robust performances.

The relationship between the L 2 -norms of z(k) and γ 2 w(k) is shown in figure 6. As clearly seen from this picture, the design objectives given by ( 38) are globally satisfied ( z(k) 2 2 < γ 2 w(k) 2 2 ). Note that the proposed objectives are not well satisfied around the time origin due to the difference between the initial conditions of the multiple model and the MIO (clearly z(0) = 0).

On the other hand, it can be noted from figure 5 that the UIs are well estimated even if the influence of disturbances on the UI estimation errors is not taken into consideration in the observer design procedure. However, UI estimations can be improved by using a dedicated observer for estimating independently states and UI of the system or by adding a filtering stage. It can also be pointed out that the UI estimations are well performed despite that they appear simultaneously. From a diagnosis point of view, this feature can be very interesting. Indeed, the UI estimations given by the proposed observer can be used as fault symptoms of the system for fault detection, isolation and identification.

5.2

State and UI estimation when the (q + 1) th -difference of the UI is not null: problem 3

In this section, faults η a (k) and η s (k) are defined by:

η a (k) =        0 if 0 ≤ kT s ≤ 2 -0.1250k + 0.25 + 0.2 cos(4k) otherwise (57) η s (k) =                0 if 0 ≤ kT s ≤ 2.5 0.2 + 0.1 sin(3k) if 2.5 ≤ kT s ≤ 7.5 0 otherwise . (58)
The MIO comprises two integral actions. However, the UI decoupling cannot be accomplished as in the previous example because the UIs are not a strict polynomial of degree one. Here, only the L 2 -gain between the disturbances and the state estimation error will be attenuated in the observer design procedure then H = [I (5×5) 0 (5×4) ]

and the weight matrix Q = I (4×4) .

Conditions of theorem 3 are fulfilled with:

K =    0
with a minimal attenuation level given by γ = 5.4772.

On figure 7 the state estimation errors provided by the proposed observer are plotted. Let us notice that these estimation errors remain globally bounded and close to zero. The comparison of the UIs and their estimates are displayed on the figure 8. As clearly seen, the proposed observer provides good dynamic and robust performances even if a total decoupling of the UI cannot be accomplished.

Conclusion

In this paper, based on a decoupled multiple model representation of a nonlinear system, the design of a multiintegral unknown input observer has been proposed. The suggested multiple model is a promising alternative to the classic structures currently employed in the multiple model approach. Indeed, the dimension of each submodel can be different and some flexibility can be expected in black box modelling of complex system. The proposed observer is an extension of the well-known multi-integral observer that is able to provide both state and unknown input robust estimation. The effectiveness of the proposed approach is illustrated via a simulation example. The suggested observer can be used, as an extension of the classic generalized observer scheme, for the detection, isolation and identification of faults acting on the systems. Improvements to the proposed observer, in order to take into consideration a more general class of UI, provides promising prospects in the future. η0 (k)

η1 (k) ηi (k) ηq-1 (k) ηq (k) K 0 K 1 K i K q-1 K q K P z -1 z -1 z -1

Figure 1 : 2 Figure 2 : 2 timeFigure 3 : 5 timeFigure 4 :Figure 5 :Figure 6 : 5 timeFigure 7 :Figure 8 :

 122235456578 Figure 1: Architecture of the unknown input MPI observer

Acknowledgements

This work is partially supported by the Conseil Régional de Lorraine (France).


Emilie Layre
email: emilie.layre@ipbs.fr

Targeted lipidomics of mycobacterial lipids and glycolipids

Keywords: Mycobacterium, lipidomics, mass spectrometry, HPLC-MS 2

Decades of study have highlighted the richness and uniqueness of the repertoire of lipid and glycolipid families produced by mycobacteria. Many of these families potently regulate host immune responses, in stimulatory or suppressive ways. Thus the global study of this repertoire in different genetic backgrounds or under model conditions of infection is gaining interest. Despite the difficulties associated with the specificities of this repertoire, the field of mass spectrometry-based lipidomics of mycobacteria has recently made considerable progress, particularly at the analytical level. There is still considerable scope for further progress, particularly with regard to the development of an efficient bioinfomatics pipeline for the analysis of the large datasets generated. This chapter describes an HPLC-MS methodology allowing the simultaneous screening of more than 20 of the lipid families produced by mycobacteria and provides recommendations to analyse the generated data given the state of the art.

Introduction

Different analytical techniques are used to analyse lipids and glycolipids including thin layer chromatography (TLC), nuclear magnetic resonance (NMR) and mass spectrometry (MS), which have their strengths and weaknesses, as previously reviewed [START_REF] Jurowski | Analytical Techniques in Lipidomics: State of the Art[END_REF]. Nevertheless, if by lipidomic we mean the rapid and "global" analysis of the lipids and glycolipids that characterise a given system, mass spectrometry appears to be the method of choice because it provides sensitivity, resolution, high throughput and allows assessing the acylforms diversity of the lipid families.

Lipidomics approaches, which have benefited from the enormous progress of mass spectrometry, are today diverse and highly developed, in particular for the analysis of eukaryotic lipids [START_REF] Zullig | Lipidomics from sample preparation to data analysis: a primer[END_REF]. These approaches share main steps: i) sample preparation (lipid extraction), ii) analytical measurement (MS-based detection), iii) data processing (lipid identification and quantification) and iv) data interpretation, which depend on biological questions and rely on more or less complex bioinformatics pipelines. The type of samples (biological fluids, tissue or cell culture) and the "targeted" versus "untargeted" nature of the study condition these different steps. Whether the study focuses on certain lipid families of the lipidome (targeted study) or, on the contrary, on the lipidome in a global way with the possible discovery of new lipids (untargeted study), will lead to favour specific methods for lipid extraction and MS devices according to their performance in terms of sensitivity, dynamic range, and resolution.

Mycobacteria produce a very diverse and specific repertoire of alkyl chain-containing molecules including lipids and glycolipids [3], lipoproteins [START_REF] Becker | Mycobacterium tuberculosis lipoproteins in virulence and immunity -fighting with a double-edged sword[END_REF] and lipoglycans, which are potent immunomodulators with key roles in host-pathogen interactions [START_REF] Neyrolles | Recent advances in deciphering the contribution of Mycobacterium tuberculosis lipids to pathogenesis[END_REF][START_REF] Ray | Bacterial cell wall macroamphiphiles: pathogen-/microbe-associated molecular patterns detected by mammalian innate immune system[END_REF]. These molecules distribute differently within the two membranes of the mycobacteria's envelope [START_REF] Chiaradia | Dissecting the mycobacterial cell envelope and defining the composition of the native mycomembrane[END_REF][START_REF] Hoffmann | Disclosure of the mycobacterial outer membrane: cryo-electron tomography and vitreous sections reveal the lipid bilayer structure[END_REF][START_REF] Zuber | Direct visualization of the outer membrane of mycobacteria and corynebacteria in their native state[END_REF][START_REF] Bansal-Mutalik | Mycobacterial outer membrane is a lipid bilayer and the inner membrane is unusually rich in diacyl phosphatidylinositol dimannosides[END_REF] (Fig. 1). The plasma membrane is mainly composed of phospholipids, phosphatidyl-myo-inositol di-and hexamannosylated (PIM2 and PIM6), triglycerides (TAG), lipoproteins, and lipoarabinomanan (LAM). The outer membrane, or mycomembrane, is made of an inner leaflet composed of the mycolic acids (MA) (up to 90 carbon-long fatty acids) that are covalently bound to the arabinogalactan, itself linked to the peptidoglycan. In contrast, the outer leaflet of the mycomembrane is not covalently bound to the inner leaflet and encloses, besides the LAM and lipoproteins, an impressive diversity of unique lipid and glycolipid families as illustrated by a few examples in Figure 1. Among these surface lipidic components, mycolyl esters such as trehalose mono-and dimycolates (TMM and TDM) are found for all mycobacteria, with variation in their mycolic acyl groups (alpha, methoxy or keto mycolic acids). In contrast, phthiocerol dimycocerosate (PDIM), phenolic glycolipids (PGL) or sulfoglycolipids (SGL) are strain-specific lipid families. Each of these families exists as acylform variants, which number can reach several dozens. Finally, the bacillus surface is coated with a detergent-labile capsule layer essentially made of glycans and, to a lower extent, lipids and proteins [START_REF] Sani | Direct visualization by cryo-EM of the mycobacterial capsular layer: a labile structure containing ESX-1-secreted proteins[END_REF]. Hence, similar to eukaryotes, mycobacteria also produce a repertoire of lipids that contains hundreds of molecular species [START_REF] Layre | A comparative lipidomics platform for chemotaxonomic analysis of Mycobacterium tuberculosis[END_REF].

Lipoproteins and lipoglycans, which are amphiphile macromolecules rather than lipids per se, are studied using specific extraction and analytical methods [START_REF] Becker | Lipoprotein Glycosylation by Protein-O-Mannosyltransferase (MAB_1122c) Contributes to Low Cell Envelope Permeability and Antibiotic Resistance of Mycobacterium abscessus[END_REF][START_REF] Laneelle | Lipid and lipoarabinomannan isolation and characterization[END_REF][START_REF] Parra | Scrutiny of Mycobacterium tuberculosis 19 kDa antigen proteoforms provides new insights in the lipoglycoprotein biogenesis paradigm[END_REF][START_REF] Nigou | Structural analysis of mycobacterial lipoglycans[END_REF]. Hence, we will focus here on the so-called "extractible" mycobacterial lipids and glycolipids that are recovered by exposing mycobacteria pellets to organic solvents. This includes lipids from the cytoplasm, plasma membrane and outer leaflet of the mycomembrane. It is important to note that the mycolic acids of the mycomembrane's inner leaflet are not included in the extractible lipid pool. Indeed, in the absence of saponification to cleave their linkage with the arabinogalactan, they remain associated to the insoluble material after exposure of mycobacteria with organic solvents.

The purification and analysis by TLC of many of these lipid families have been widely described [START_REF] Slayden | Analysis of the Lipids of Mycobacterium tuberculosis[END_REF] and is also detailed in previous editions of this protocol book [START_REF] Lanéelle | Lipid and Lipoarabinomannan Isolation and Characterization[END_REF]. Mass spectrometry-based detection methods relying on the use of MALDI-Tof or ESI/APCI-MS instruments have also been used for the analysis of MA [START_REF] Portevin | Lipidomics and genomics of Mycobacterium tuberculosis reveal lineage-specific trends in mycolic acid biosynthesis[END_REF][START_REF] Shui | Mycolic acids as diagnostic markers for tuberculosis case detection in humans and drug efficacy in mice[END_REF], SGL [START_REF] Layre | Deciphering sulfoglycolipids of Mycobacterium tuberculosis[END_REF][START_REF] Jain | Lipidomics reveals control of Mycobacterium tuberculosis virulence lipids via metabolic coupling[END_REF][START_REF] Larrouy-Maumus | Mycobacterial envelope lipids fingerprint from direct MALDI-TOF MS analysis of intact bacilli[END_REF], DIM [START_REF] Jain | Lipidomics reveals control of Mycobacterium tuberculosis virulence lipids via metabolic coupling[END_REF], PIMs [START_REF] Gilleron | MS/MS approach for characterization of the fatty acid distribution on mycobacterial phosphatidyl-myo-inositol mannosides[END_REF][START_REF] Hsu | Structural characterization of phosphatidyl-myo-inositol mannosides from Mycobacterium bovis Bacillus Calmette Guerin by multiple-stage quadrupole ion-trap mass spectrometry with electrospray ionization. I. PIMs and lyso-PIMs[END_REF][START_REF] Hsu | Structural characterization of phosphatidyl-myo-inositol mannosides from Mycobacterium bovis Bacillus Calmette Guerin by multiple-stage quadrupole ion-trap mass spectrometry with electrospray ionization. II. Monoacyl-and diacyl-PIMs[END_REF][START_REF] Larrouy-Maumus | Mycobacterial envelope lipids fingerprint from direct MALDI-TOF MS analysis of intact bacilli[END_REF], PGL [START_REF] Larrouy-Maumus | Mycobacterial envelope lipids fingerprint from direct MALDI-TOF MS analysis of intact bacilli[END_REF], TMM [START_REF] Fujita | Direct molecular mass determination of trehalose monomycolate from 11 species of mycobacteria by MALDI-TOF mass spectrometry[END_REF] ,TDM [START_REF] Fujita | Intact molecular characterization of cord factor (trehalose 6,6'-dimycolate) from nine species of mycobacteria by MALDI-TOF mass spectrometry[END_REF] and siderophores (mycobactin and carboxymycobactin families) [START_REF] Madigan | Lipidomic discovery of deoxysiderophores reveals a revised mycobactin biosynthesis pathway in Mycobacterium tuberculosis[END_REF] and might be favoured for the specific high throughput analysis of one of these lipid family. More recently, lipidomic methods have been developed to analyse the M. tuberculosis lipid repertoire in a more global way. This effort has faced the challenges associated with the unique and structurally diverse mycobacterial lipids, which prevented the direct application of analytical methods developed for eukaryotic lipids. Two main studies have described HPLC-(ESI)-Q-Tof approaches that differ in the reverse phase (C18) [START_REF] Sartain | Lipidomic analyses of Mycobacterium tuberculosis based on accurate mass measurements and the novel "Mtb LipidDB[END_REF] versus normal phase (diol) [START_REF] Layre | A comparative lipidomics platform for chemotaxonomic analysis of Mycobacterium tuberculosis[END_REF] chromatographic method. While the first one separates lipids based on their acyl groups (by acylform), the second one separates lipids based on their polar head (by family), but both approaches provide a wide coverage of M. tuberculosis lipidome. Both studies also released a theoretical database listing the calculated m/z of adducts of the published M. tuberculosis lipid families, called Mtb LipidDB (https://mrl.colostate.edu/mtb/) [START_REF] Sartain | Lipidomic analyses of Mycobacterium tuberculosis based on accurate mass measurements and the novel "Mtb LipidDB[END_REF] and MycoMass (https://www.brighamandwomens.org/research/departments/rheumatology-immunologyallergy/moody-lab/overview) [START_REF] Layre | A comparative lipidomics platform for chemotaxonomic analysis of Mycobacterium tuberculosis[END_REF], respectively.

If the M. tuberculosis lipidomic field has made significant progress it lags behind the lipidomics of eukaryotic lipids because it still suffers from several shortcomings: the lack of a MS/MS database that would allow Multiple Reaction Monitoring mode analysis for the identification of lipids, the poor availability of standards to achieve absolute quantification, and the lack of a straightforward bioinformatics pipeline for high-throughput data analysis, especially for unbiased and comparative analysis of large number of datasets.

Based on the previously published method that relies on diol chromatography coupled to (ESI)-MS [START_REF] Layre | A comparative lipidomics platform for chemotaxonomic analysis of Mycobacterium tuberculosis[END_REF], this chapter describes a lipidomic approach to perform the simultaneous screening of more than 20 of the main M. tuberculosis lipid families. This method has been successfully used in the past to characterise M. tuberculosis lipid profile in response to stress [START_REF] Galagan | The Mycobacterium tuberculosis regulatory network and hypoxia[END_REF], to compare the lipid profile of mycobacteria strains [START_REF] Layre | Molecular profiling of Mycobacterium tuberculosis identifies tuberculosinyl nucleoside products of the virulence-associated enzyme Rv3378c[END_REF] and also to detect mycobacterial lipid in tissues [START_REF] Layre | Molecular profiling of Mycobacterium tuberculosis identifies tuberculosinyl nucleoside products of the virulence-associated enzyme Rv3378c[END_REF]. The mass spectrometer available in the environment of a given laboratory is a limiting and inflexible factor, given the cost of such an instrument. Thus, the parts of this chapter describing the analytical detection of mycobacterial lipids and the data analysis correspond to recommendations to implement this approach and ensure quality of the analyses rather than a wet-lab protocol. 5. High-resolution mass spectrometer (see Note 3).

Material

Targeted screening and comparative analysis

1. R software 2. msConvert ProteoWizard 3. MS database: Mtb LipidDB [START_REF] Sartain | Lipidomic analyses of Mycobacterium tuberculosis based on accurate mass measurements and the novel "Mtb LipidDB[END_REF] (https://mrl.colostate.edu/mtb/) or MycoMass [START_REF] Layre | A comparative lipidomics platform for chemotaxonomic analysis of Mycobacterium tuberculosis[END_REF] (https://www.brighamandwomens.org/research/departments/rheumatology-immunology-allergy/moody-lab/overview)

Methods

Glycolipid extraction from mycobacteria culture

1.

Transfer 50 mL of mycobacterial culture grown in the absence of detergent (see Note 4) into 50 mL polypropylene V-bottom tubes.

2.

Centrifuge the tubes at 2000 g for 15 min at room temperature and discard the supernatant.

3.

Wash the pellet twice with 10 mL of water by resuspending the bacteria and repeating step 2.

4.

Resuspend the bacterial pellet in 1 mL of methanol.

5.

Add internal lipid standard (see Note 5).

6.

Transfer the methanolic bacterial suspension to 50 mL screw centrifuge glass tubes with PTFE coated screw caps containing 45 mL of 1:2 chloroform:methanol (v :v) (see Note 6).

7.

Allow sterilisation overnight (see Note 7).

8.

Shake tubes on a rocking shaker for about 1 h.

9.

Pellet the bacteria at 2000 g for 15 min (or until a translucent supernatant is obtained), at room temperature.

10.

Transfer and dry the supernatant in clean 250 mL-round bottom flask or in 15 mL-glass tubes depending on available evaporating system.

11.

Resuspend the pellets in 45 mL of 1:1 chloroform:methanol (v:v) and repeat steps 8-10. Pool the organic supernatants.

12.

Resuspend the pellets in 45 mL of 2:1 chloroform:methanol (v:v) and repeat steps 8-10. Pool the organic supernatants.

13.

Dry the pooled organic supernatants using a centrifugal evaporator, gene-vac or under nitrogen stream in a chemical hood.

14.

Resuspend the dried lipids in 1 mL of 1:1 chloroform:methanol (v:v), transfer to a pre-weighted 2 mL-amber glass vial with a Hamilton syringe and dry under a gentle nitrogen stream without heat.

15.

Rinse with additional 1 mL of 1:1 chloroform:methanol (v:v), add to the 2 mL-amber glass vial with a Hamilton syringe and dry under a gentle nitrogen stream without heat.

16.

Weight the dry lipids (see Note 8), resuspend at 2 mg/mL in 1:1 chloroform:methanol (v:v), store at -80°C if not analysed immediately (see Notes 9).

HPLC-MS analysis

3.2.a Set-up of the HPLC-MS system for targeted screening of mycobacterial lipids

1.

Equilibrate the HPLC Diol column at room temperature with Solvent A at 0.15 mL/min for 30 min (see Note 10).

2.

Establish the (m/z, RT) map of the lipid families of interest using your HPLC-MS instrument (see Note 11 The lipids are detected at m/z matching the calculated m/z within the mass accuracy of the high-resolution mass spectrometer (as low as 1 ppm and never exceeding 10 ppm mass accuracy) and show the expected acylform profile when analysing purified lipid family or total lipid extract (see Note 14).

Observed retention time matches the retention time of standard (when available), i.e. synthetic or purified acylform of the lipid family (see

Note 12).

The fragmentation pattern obtained by MS/MS confirms the building blocks that characterise the lipid family (see Note 15 ).

Table 1 lists the characteristics of the MS signal of the lipid families detected analysing the M. tuberculosis H37Rv lipid extract using the Diol HPLC-Agilent 6520 Accurate-Mass Q-Tof.

3.

Ensure reproducibility and stability over time of the assigned MS signals by performing multiple and consecutive analyses of the same M. tuberculosis lipid extract. Alternate with blank runs (injection of 1:1 chloroform:methanol (v:v) or solvent A) to confirm absence of detection of carryover during the blank runs.

4.

Store aliquots of the mycobacterial lipid mixture (purified/synthetic mycobacterial lipids or natural lipid extract) used for parameters optimisation and the associated datasets to control the response of the mass spectrometer before a new experiment and after mass spectrometer maintenance.

3.2.b Perform targeted screening of mycobacterial lipids in lipid extract of interest

1.

Equilibrate the HPLC-MS diol column at room temperature with Solvent A at 0.15 mL/min for 30 min (see Note 10).

2.

Ensure the cleanliness and the proper functioning of the system by analysing an alternation of blank samples (solvent used to solubilise the samples, see subheading 3.1 step 14) and lipid reference samples using the gradient described above (subheading 3.2.a, step 2).

3.

Inject 10-20 µg of the lipid extract of interest resuspended in 10-20 µL of chloroform:methanol 1:1 (v:v) or solvent A (see Note 13). If needed, centrifuge samples at 2000 rpm for 5 min to remove potential insoluble residues before injection. 4. If semi-quantitative comparative analysis is the goal, determine the quantity of lipid extract to be injected for a linear response of the different lipid families to be screened and compared (see Note 16):

-Inject a range from 0 to 20 µg of reference lipid extract.

-Extract ion chromatograms for each lipid of interest and integrate area under the chromatograms.

-Plot the curve of chromatogram area versus injected quantity of sample for the different lipid families of interest.

-Determine the injection quantity range over which the majority of lipids show a linear response.

3.3

Targeted screening and comparative analysis

• Semi-automated screening of mycobacterial lipid families

Once the map of detected lipids is established routine screening of these families can be undertaken. For a moderate number of analyses, one of the easiest ways is to use the data acquisition and visualisation software associated to the spectrometer and to extract the ion chromatograms at the expected m/z for representative or all acylforms of the lipid families (see Note 14), with a tolerance within the mass spectrometer mass accuracy (1-10 ppm). It is often possible to pre-register a list of m/z for the simultaneous extraction of the ion chromatograms of all the lipids of interest.

• Automated lipid annotation

For a large number of samples such as biological fluids from patient cohorts or a mycobacteria library, it can be envisaged to set up a bioinformatic workflow for the extraction of all the chromatograms of the generated datasets using an appropriate peak picking algorithm and/or to carry out an automatic annotation with software integrating the available databases, MycoMass [START_REF] Layre | A comparative lipidomics platform for chemotaxonomic analysis of Mycobacterium tuberculosis[END_REF] or Mtb LipidDB [START_REF] Sartain | Lipidomic analyses of Mycobacterium tuberculosis based on accurate mass measurements and the novel "Mtb LipidDB[END_REF] (see Table 2 and Note 17), such as the MS-LAMP software [START_REF] Sabareesh | Mass spectrometry based lipid(ome) analyzer and molecular platform: a new software to interpret and analyze electrospray and/or matrix-assisted laser desorption/ionization mass spectrometric data of lipids: a case study from Mycobacterium tuberculosis[END_REF].

Several lipidomic data processing software packages have been developed for filtering, peak picking, peak integration and alignment, such as the online XCMS platform (https://xcmsonline.scripps.edu) [START_REF] Smith | XCMS: processing mass spectrometry data for metabolite profiling using nonlinear peak alignment, matching, and identification[END_REF], MZmine 2 [START_REF] Pluskal | MZmine 2: modular framework for processing, visualizing, and analyzing mass spectrometry-based molecular profile data[END_REF] or MS-DIAL [START_REF] Tsugawa | MS-DIAL: data-independent MS/MS deconvolution for comprehensive metabolome analysis[END_REF]. Each require converting the raw MS files in mzXML format, which can be performed either with the data export module of the mass spectrometer data acquisition software (such as Agilent MassHunter) or can be carried out using msConvert ProteoWizard [START_REF] Adusumilli | Data Conversion with ProteoWizard msConvert[END_REF][START_REF] Kessner | ProteoWizard: open source software for rapid proteomics tools development[END_REF]. Processed data can then be exported toward data annotation software.

• Semi-quantitative comparison of mycobacterial lipid families present in studied samples

The area under the extracted-ion chromatogram of a given lipid species correlates with its abundance. However, translating this area in absolute quantity requires performing dose response curves using standards, which is still challenging for mycobacterial lipids due to the poor availability of standards. As well, differences in chromatograms areas between lipid classes cannot be translated into differences in absolute quantity without determining the response factor of each lipid class. Nonetheless, changes in the area under the chromatogram for a given lipid across samples of the same nature reflects changes in absolute quantity. Hence, the comparison of the lipid profiles obtained for different samples of the same nature allows highlighting changes in the relative proportion of the different lipid families without knowing their absolute value.

Normalization of the experiment is crucial to perform such semi-quantitative comparative analysis. Since the use of detergent in mycobacteria culture is not recommended, strong bacterial aggregation is observed and optical density of the cultures cannot be used for the normalisation of the comparative analysis of different mycobacterial strains (see Notes 4). Instead, comparative analysis can be normalized based on the weight of extracted bacteria pellet or on the amount of injected lipid extract (assuming that the same amount of lipids are extracted per bacillus for different strains) (see Note 8).

These considerations being said, the comparative analyses of samples should be performed as follows:

1-Add an internal standard(s) (see Note 5) to studied samples before lipid extraction to control the quality of the extraction process and normalisation.

2-Perform HPLC-MS analysis as above described by injecting the amount of lipid extract according to the chosen normalisation method (see Notes 8 and 18) and to the previously determined range of linear response of the studied lipid families (see Note 16).

3-Extract for each dataset the ion chromatograms at the expected m/z for representative or all acylforms of the lipid families (see Note 14), with a mass tolerance within the mass spectrometer mass accuracy (1-10 ppm), and integrate signals under the chromatogram curves.

Notes

1.

Handle all organic solvents under the hood. Chloroform can be replaced by dichloromethane, which is less carcinogenic.

2.

This approach was initially developed using an Agilent Technologies 6520 Accurate-Mass Q-Tof and a 1200 series HPLC system with a Varian MonoChrom Diol (dihydroxypropyl group bonded phase) column (150 mm × 2 mm, 3 μm particles of 10Å porosity) and a Varian MonoChrom Diol guard column (3 μm × 4.6 mm). Elution profile obtained with the Varian Diol column is presented in Table 1. This column is discontinued but equivalent Diol columns with the same characteristics are available (for example, Agilent or Princeton diol columns). Using equivalent Diol columns leads to similar order of elution of M. tuberculosis lipids families but with possible moderate variation in retention time.

3.

Mass spectrometers show variable performance (sensitivity and dynamic range) and M. tuberculosis lipid families have different ionization properties. Hence, applying this approach to a different mass spectrometer might require the optimization of the ionization parameters of the instrument in order to allow the detection of the different lipid families. For information, this approach has been later successfully transferred on LTQ Orbitrap XL Hybrid Ion Trap-Orbitrap Mass Spectrometer (Thermo) mass spectrometer coupled to a Dionex Ultimate 3000 HPLC system and to a Xevo G2 Qtof (WATERS) mass spectrometers (unpublished).

4.

Traces of detergents alter lipidomic analyses by inducing important contaminant mass spectrometry signals and by suppressing the signal of relevant lipids. It is therefore necessary to take several precautions with regard to the culture media, the quality of the solvents and the appropriate use of plastic and glass materials.

Culture conditions: Mycobacteria are usually grown in standard Middlebrook 7H9 (Difco) or Sauton medium. It is recommended to avoid the use of detergent such as Tween, which is routinely employed to reduce the characteristic aggregation of mycobacteria in liquid media. Indeed, detergents stick to the bacillus envelope and are recovered during lipid extraction by organic solvents. When growth monitoring by optical density measurement is required, it is preferable to use tyloxapol. Another alternative is to prepare a replicate culture in the presence of detergent in parallel with detergent-free cultures for each strain, in order to estimate their growth phase based on optical density.

Material: Material is also a source of contaminations. Methanol is compatible with plastic consumables but as soon as chloroform-or dichloromethane is used any contact with plastic consumables (pipette, tubes and caps) is strictly forbidden and therefore glass material must be used (pipette, syringe and tubes with PTFE caps). In addition, glassware washed with soap such as in common laboratory laundry is still a source of contaminations. Hence, attention has to be paid to clean the glassware twice with 1:1 chloroform:methanol before use. Polytetrafluoroethylene (PTFE) caps must be used with screw glass tubes.

Solvents: All solvents should be HPLC-grade and freshly prepared.

5.

Addition of a known quantity of non-mycobacterial lipid standard, such as phosphatidylcholine for example, to the bacteria pellet allows controlling the overall lipid extraction process and to normalize comparative analyses. The intensity of the MS signal of the chosen standard should be within the intensity range of the analytes of interest. The determination of the response factor of the standard in the used LC-MS set up will determine the suitable amount of standard to be added.

6.

The tip of the plastic pipette must not contact the chloroform:methanol solution when transferring the methanolic bacteria solution. All following steps of extraction protocol must be performed in a chemical hood 7.

When manipulating live virulent M. tuberculosis strains, step 1-7 are performed in BSL3 facility. Contacting the bacterial pellet with chloroform:methanol mixture also aims at killing the bacteria before pursuing the protocol outside of the BSL3 facility. Successful sterilization relies on the ratio wet pellet / chloroform:methanol. From 50 mL of mycobacterium culture, recovered wet pellet is typically ≤ 1 mL. A ratio of 4 parts solvent to 1 part wet pellet (v/v) is expected to sterilize the sample overnight. In any case, when setting up this protocol, the sterilization conditions must be validated by a controlled experiment showing the absence of colony forming unit when plating the so-called "sterile" pellet on Petri dish. Once the sample is sterilized, spray the outside of the glass container with tuberculocidal agent and continue the protocol outside of the BSL3 facility in a chemical hood using glass pipettes or syringes and rinsed glassware as described in Note 4.

8.

About 1-1.5 mg of dried lipids is usually recovered from 50 mL of log phase culture. If normalization of experiments is later carried out on the quantity of injected lipid extract, weight measurement must be as reliable as possible, which requires the complete evaporation of the solvents.

Repeat weighing of the dried lipid film on a microbalance 20 min later and ensure a constant weight is reached otherwise desiccation time must be prolonged. Drying under vacuum might be necessary depending on the amount of lipid extract.

9.

Best results are obtained with freshly prepared extracts. Lipid extracts should not be stored in a dry state. Oxidative degradation is slower in solutions (up to 1 year). Dichloromethane is more stable than chloroform for long-term storage, which should take place in a flammable storage laboratory freezer. Air and light should be avoided. To minimise the cleavage of ester lipids during long periods of storage, methanol can be replaced by 2-propanol in the solvent mixture.

10.

HPLC solvents A and B should be freshly prepared and should not be stored. If (light) precipitates are observed when adding the additives to the solvent mixtures, 1 minute in a sonic bath can help the solubilisation. The HPLC system should be equipped with solvent resistant gaskets and solvent bottles with integrated air valve to avoid solvent evaporation. Sufficient volume of solvents should be prepared in order to analyse all the samples to be compared with the same batch of solvents and to avoid retention time variation.

11.

For future screening of mycobacterial lipids in different samples or for different mycobacteria strains it is necessary to first establish a detection map (m/z, retention time) of these different families for a given HPLC-MS system. Mass spectrometers show variable performance (sensitivity and dynamic range) and M. tuberculosis glycolipid families have different ionization properties. Hence, applying this approach with a given mass spectrometer might require the optimization of the ionization parameters of the instrument in order to allow the detection of all lipid families. For information, initially set up using an Agilent 6520 (ESI)-Q-tof mass spectrometer, this approach has been later successfully transferred on a Thermo LTQ Orbitrap XL and a Waters Xevo G2 Qtof mass spectrometers (unpublished). Using the Agilent Technologies 6520 Accurate-Mass Q-Tof in full scan mode, recommended Dual ESI source and acquisition parameters are the following, as previously published [START_REF] Jurowski | Analytical Techniques in Lipidomics: State of the Art[END_REF]: gas temperature 325°C, 5 L/min drying gas flow, 30 psig nebulizer pressure, VCap 5,500 V. Spectra are collected from m/z 100 to 3,000 at 1 spectrum/s with continuous infusion of calibrants (m/z 121.050873 and 922.009798 in positive ion mode, m/z 112.985587 and 1033.98810 in negative ion mode).

12.

Standards of mycobacterial lipids are currently poorly marketed. Polar Aventi mycolic acids are distributed by Sigma Aldrich and " Mycobacterium tuberculosis, Strain H37Rv, Total Lipids" (NR-14837) is provided by the BEI resources. Alternatives for MS signal optimisation are to directly work with the total lipid extract from M. tuberculosis culture to reproduce the lipid profile detailed in Table 1 or to contact laboratories with expertise in M. tuberculosis lipids purification or organic synthesis to obtain pure lipids for the optimization of individual lipid families detection.

13.

The amount of lipids to be injected depends on the instrument sensibility. Agilent Technologies 6520 Accurate-Mass Q-Tof requires about 10 micrograms of M. tuberculosis total lipid extract to detect the different lipid families listed in Table 1. The same instrument shows a limit of detection of 10 -12 to 10 -9 mole injected when analysing purified lipid standards [START_REF] Jurowski | Analytical Techniques in Lipidomics: State of the Art[END_REF].

14.

Each M. tuberculosis lipid family is composed of not one, but more often of several dozen acylforms that share a common polar head.

Hence, the signal corresponding to a given lipid family is characterized by a series of molecular ions detected with m/z values that differ by 14.0156 a.m.u (CH2 group), which elute within a narrow retention time window on diol chromatography. When performing the semi-quantitative comparison of the lipid profile of different samples, it is possible to perform the comparison using one representative acylform by lipid family, if the acylform profile of the lipid families is conserved between samples.

15.

While MS/MS databases are not available for M. tuberculosis lipids, fragmentation profiles of several M. tuberculosis lipids families have been previously published [START_REF] Jurowski | Analytical Techniques in Lipidomics: State of the Art[END_REF].

16.

When the goal is the semi-quantitative comparison of the lipid profiles of different strains (for example, WT, mutant and complemented strains) or of a same strain grown in different conditions, it is important to ensure that the injected quantity of lipid extract is within the linear response range of the lipids of interest. Indeed, the area under the chromatogram of a given molecular lipid species increases proportionally with its abundance until the signal saturates or even gradually extinguishes. Because each lipid family has its own response factor under given mass spectrometry settings, it is therefore important that the concentration of the studied lipid extract (i.e. the injected quantity) is suitable to assess changes in abundance for as many, if not all, lipid families as possible.

17.

These databases represent a unique resource of information but it is important to keep in mind that they consist of lists of theoretical calculated m/z for the acylforms of the different mycobacterial lipid families described in the literature. The MycoMass was built to provide a tool to assist MS signals assignments or at least to propose possible formulas. As culture/infection conditions modulate the acylform profile of lipids, the list of listed acylforms is not limited to the published data but is sometimes extended in terms of acyl chain length and combination of fatty acids (Table 2). In addition, these databases do not contain retention time criteria, which are method-and therefore laboratory-dependent. Hence, a match between a detected m/z and an entry in these databases does not constitute, by itself, a definitive assignment. The use of these databases for the annotation of large HPLC-MS datasets currently requires an extensive manual curating to remove annotations that are incoherent with observed RTs. When using these databases on datasets generated using a different chromatographic separation, annotations of MS signals must be confirmed by MS/MS experiments. The time saved using database for the annotation of large MS datasets is to the user's appreciation.

18.

When comparing the lipid profile of different strains, it is desirable to ensure that the cultures are in the same growth phase (log versus late phase). One replicate culture in the presence of detergent can be carried out in parallel with detergent-free cultures for each strain, in order to estimate their growth phase based on optical density measurement. Coverage of the repertoire of M. tuberculosis lipids will depend on the mass spectrometer's performances. It is to note that TDM is absent of the list.

5.

Figure/Table Legends

TDM produced by Mycobacterium species is a family of very large (m/z centered ~ 2700 a.m.u) and hydrophobic acylforms, which are poorly ionising by electrospray. Therefore, TDM cannot be listed as routinely detected by HPLC-MS. TLC and MALDI-Tof are currently the most successful methods for TDM detection [START_REF] Fujita | Intact molecular characterization of cord factor (trehalose 6,6'-dimycolate) from nine species of mycobacteria by MALDI-TOF mass spectrometry[END_REF].

  ): Inject purified/synthetic M. tuberculosis lipids (see Note 12) or reference M. tuberculosis lipid extract and apply the following gradient at room temperature and at a flow rate of 0.150 mL/min as follows (see Note 13) : Time (min) A/B (%) M. tuberculosis lipids MS signals in your system. In the absence of MS/MS database, the assignment of MS signals to given lipid families relies on the validation of at least 2 of following criteria:

Figure 1 :

 1 Figure 1:Examples of unique lipids extracted by organic solvents from the mycobacterial envelope. Exposure of mycobacteria with organic solvents allows recovering the "extractable lipids" that compose the plasma membrane and the mycomembrane. AG: arabinogalactan, LAM: lipoarabinomannan, LM: lipomannan, LP: lipoprotein, MA: mycolic acids.

2.1 Glycolipid extraction from mycobacteria culture

	9. 500 µL Hamilton syringes
	10. Pasteur glass pipettes
	11. Rocking shaker
	12. Nitrogen stream drying system
	13. Microbalance
	14. 2 mL-amber vials with PTFE screw caps
	2.2 HPLC-MS analysis
	1. LC-MS-grade hexane, 2-isopropanol, methanol, formic acid (CH3COOH) and ammonium hydroxide (NH4OH)
	2. Solvent A: 70:30 hexane : isopropanol (v :v), 0.1% (v/v) formic acid, 0.05% (v/v) Ammonium hydroxide
	3. Solvent B: 30:70 methanol : isopropanol, 0.1% (v/v) formic acid, 0.05% (v/v) Ammonium hydroxide
	4. Diol (dihydroxypropyl group bonded phase) HPLC column (150 mm × 2 mm, 3 μm particles of 10 Å porosity) and a Diol guard column (3
	μm × 4.6 mm) (see Note 2).
	1. Chemical fume hood
	2. LC-MS-grade chloroform (CHCl3) or dichloromethane (CH2Cl2) (see Note 1)
	3. LC-MS-grade methanol (CH3OH)
	4. LC-MS-grade water.
	5. Solvent evaporation system: centrifugal evaporator, Genevac or nitrogen stream
	6. 50 mL polypropylene V-bottom tubes (Falcon type)
	7. 50 mL screw centrifuge glass tubes (Pyrex quality) with PTFE coated screw caps
	8. 250 mL round bottom flasks if using rotary vacuum evaporator as drying system or 15 mL-glass tubes if using stream-drying system to dry
	organic phases.

Table 1 :

 1 List of detected mycobacterial lipid families and acylforms. Are listed the formula of the molecular ions of each acylform, the detected m/z of the indicated adduct, and their indicative retention times (subjected to changes depending on the Diol column brand). Detected m/z are listed for only one ion-mode. In positive ion-mode all lipid families are detected and [M+NH4] + and [M+H] + adducts are the most commonly detected adducts.In negative ion mode, detected lipids are those that are naturally negatively charged [M-H] -such as the phosphated and sulphated lipids (phospholipids, PIM and sulfoglycolipids) while neutral glycolipids are either not detected or detected as [M+HCOO -] -adducts depending on the MS instrument.

Table 2 :

 2 Excerpt of the MycoMass database illustrating the provided data and their organisation, which follows the LIPIDMAPS classification.

		C81H160O4	neg/[M-H-]-	1196.2244
		C82H162O4	neg/[M-H-]-	1210.2400
		C83H164O4	neg/[M-H-]-	1224.2557
	mycolic acids	C84H166O4	neg/[M-H-]-	1238.2713	3.6-3.7
		C85H168O4	neg/[M-H-]-	1252.2870
		C87H172O4	neg/[M-H-]-	1280.3183
		C88H174O4	neg/[M-H-]-	1294.3339
		C51H98O6	pos/[M+NH4]+	824.7706
		C53H100O6	pos/[M+NH4]+	850.7864
		C54H102O6	pos/[M+NH4]+	864.8020
		C55H104O6	pos/[M+NH4]+	878.8177
		C56H108O6	pos/[M+NH4]+	894.8488
		C57H108O6	pos/[M+NH4]+	906.8490
		C57H110O6	pos/[M+NH4]+	908.8645
	triacylglycerols				3.7
		C58H112O6	pos/[M+NH4]+	922.8801
		C59H112O6	pos/[M+NH4]+	934.8803
		C60H116O6	pos/[M+NH4]+	950.9114
		C61H116O6	pos/[M+NH4]+	962.9116
		C61H118O6	pos/[M+NH4]+	964.9271
		C62H120O6	pos/[M+NH4]+	978.9427
		C63H122O6	pos/[M+NH4]+	992.9584
		C35H68O5	pos/[M+NH4]+	586.5410
		C36H70O5	pos/[M+NH4]+	600.5567
		C37H70O5	pos/[M+NH4]+	612.5567
	diacylglycerols				3.8-3.9
		C37H72O5	pos/[M+NH4]+	614.5723
		C38H74O5	pos/[M+NH4]+	628.5880
		C39H74O5	pos/[M+NH4]+	640.5880
		C56H80O2	pos/[M+H]+	785.6231
		C56H82O2	pos/[M+H]+	787.6388
	menaquinone				3.9
		C56H80O2	pos/[M+NH4]+	802.6491
		C56H82O2	pos/[M+NH4]	804.6648
		C107H198O18	pos/[M+NH4]+	1789.491649
	triglycosylated phenolic glycolipids (PGL)	C108H200O18	pos/[M+NH4]+	1803.507299	4
		C109H202O18	pos/[M+NH4]+	1817.522949

Table 1

 1