Ana Alonso Rodríguez

Francesca Rapetti
email: francesca.rapetti@univ-cotedazur.fr

On a generalization of the Lebesgue's constant

Keywords: High-order Whitney's forms, weights on subsimplices, Lebesgue's constant, Vandermonde's matrix 2020 MSC: 65N30, 65D05

In this work we generalize the definition of the Lebesgue's constant to the case of field interpolation by high-order Whitney's forms on simplices. We underline the important theoretical concepts at play.

Introduction

The polynomial interpolation of a field f over a domain Ω ⊂ R d , d ≥ 1, consists in the construction of a polynomial of degree r ≥ 0 which has a finite number of degrees of freedom (dofs) concident with those of f . This problem is a classical topic in approximation theory and matters because the high quality polynomial interpolation of fields plays an essential role in finite elements (FEs). Standard dofs for the interpolation of a scalar field f in Ω are the values of f at some chosen nodes in the domain and the quality of this approximaton is evaluated on the basis of the well-known Lebesgue's constant associated with the set of nodes (see, e.g, [START_REF] Chung | On lattices admitting unique Lagrange interpolations[END_REF], [START_REF] Pasquetti | Spectral element methods on unstructured meshes: Which interpolation points ?[END_REF]). In many areas of physics, FEs should be able to represent vector fields from a finite number of physical coefficients, the nature of which, fluxes, circulations, etc., associate them with geometric mesh objects other than nodes [START_REF] Bossavit | Computational Electromagnetism[END_REF]. In electromagnetism, for example, fields are invisible and forces act without contact. The perturbations, caused to the surrounding by the presence of the field, are observed via quantities, such as electromotive forces, intensities, etc., which correspond to weights on manifolds, namely line integrals (circulations), surface integrals (fluxes), etc., other than simply pointwise evaluations. The entity of physical significance is thus the k-form, i.e. the mapping curve→circulation (k = 1), surface→flux (k = 2), etc., and not the field itself, i.e. the electric field, the magnetic induction, etc., as a vector. The form w models the field; manifolds S are probes that one locates in Ω to measure the field. The value of the mesure, S w, is the weight of the form w on the manifold S. In FEs, these weights can be considered as dofs. They reflect the nature (and, if properly chosen, global regularity properties) of the field they represent and are the right objects to produce accurate high-order interpolation formulae.

They are the key to extend the concepts of Lebesgue's function and constant to the interpolation of any field over a domain Ω. This generalization is a novelty, resulting from adopting in FEs a point of view typical of differential geometry, thus weights as dofs.

The classical framework

Let T be a d-simplex of R d , P r (T ) the space of polynomials of degree r in T and N r = ( d+r d ) = dim(P r (T )). Given X r (T ) = {x 1 , ..., x Nr }, a set of N r distinct nodes in T , the Lagrange interpolation problem for a given scalar function f ∈ C 0 (T ) consists in finding a polynomial I r f ∈ P r (T ), such that

I r f (x i ) = f (x i ), ∀ i = 1, ..., N r . (1)
Chosen a basis B = {ψ j } Nr j=1 for the space P r (T ), the interpolation problem is well-defined if the N r × N r generalized Vandermonde matrix V B,Xr with entries (V Xr,B ) ij = ψ j (x i ), i, j = 1, ..., N r , is invertible. The name of Vandermonde matrix is given to V Xr,B in recognition of the fact that, for d = 1, if the chosen basis is the canonical one {x j-1 } Nr j=1 , then det (V Xr,B ) coincides with the Vandermonde determinant Π 1≤j<i≤Nr (x i -x j ). The solution I r f of the interpolation problem (1) can be written in terms of the basis functions in B as I r f (x) = Nr j=1 c j ψ j (x) , for x ∈ T , where the vector c = (c 1 , ..., c Nr ) is the solution of the linear algebraic system

V Xr,B c = f , f = (f (x 1 ), ..., f (x Nr )) . (2)
Inverting the matrix V Xr,B is necessary to compute the dual basis {φ Xr j } Nr j=1 ⊂ P r (T ) such that φ Xr j (x i ) = δ ij , for 1 ≤ i, j ≤ N r , being δ .. the Kronecker's symbol. The set of points X r (T ) = {x 1 , ..., x Nr } is called unisolvent for the space P r (T ) if the Lagrange interpolation problem (1) is well-defined, namely if for a given basis B of P r (T ) the Vadermonde matrix V Xr,B is invertible.

Definition 1. Given a d-simplex T and a set X r (T ) of N r points in T , unisolvent for the space P r (T ), the Lebesgue's function L Xr : T → R + is defined as

L Xr (x) = Nr j=1 |φ Xr j (x)|
with {φ Xr j } Nr j=1 ⊂ P r (T ) the Lagrange basis associated with X r (T ). The Lebesgue's constant in T is denoted by Λ Xr (T ) and defined as the maximum of the Lebesgue's function L Xr (•) over T , namely

Λ Xr (T ) = L Xr C 0 (T ) := max x∈T |L Xr (x)| .
To ensure the stability of the interpolating problem when increasing the polynomial degree, it is crucial to keep under control the growth of Λ Xr (T ) with respect to r. The Lebesgue's constant appears in the comparison between the interpolation error ||f -I r f || C 0 (T ) in the maximum norm and the best-fit error ||f -f * r || C 0 (T ) in the same norm. Indeed, for any function f ∈ C 0 (T ), we have

||f -I r f || C 0 (T ) ≤ (1 + Λ Xr )||f -f * r || C 0 (T ) (3)
where f * r is the best-fit polynomial of degree r of the function f with respect to the maximum norm in T , namely, f * r ∈ P r (T ) and

f -f * r C 0 (T ) ≤ f -p C 0 (T ) for all p ∈ P r (T ), so ||f -f * r || C 0 (T ) = inf p∈Pr(T ) ||f -p|| C 0 (T ) = inf p∈Pr(T ) max x∈T |f (x) -p(x)|.
The magnitude of the Lebesgue's constant Λ Xr (T ) depends heavily on the distribution of the points x i ∈ X [START_REF] Pasquetti | Spectral element methods on unstructured meshes: Which interpolation points ?[END_REF]. In the bound (3), the growth of the Lebesgue's constant Λ Xr determines the convergence in the maximum norm. Indeed, we have the indeterminate form lim r→+∞ ((1

+ Λ Xr )||f -f * r || C 0 (T ) ) = ∞ • 0.
If Λ Xr grows faster in r than the best-fit error dies away, convergence in r is impossible to attain. The indetermination is solved when it grows up slowly with respect to the interpolation degree r, that is for example when it satisfies the condition lim r→+∞ Λ 1/r Xr = 1, as proved in [START_REF] Bloom | The Lebesgue constant for Lagrange interpolation in the simplex[END_REF]. Moreover, the inequality (3) suggests that, when the Lebesgue's constant does not grow too fast, we can find an approximation of a function f on T that is almost as good as the best-fit f * r , by just taking the polynomial interpolant I r f , which is generally much easier to compute than f * r .

Whitney's forms

For any integer k, with 0 ≤ k ≤ d a k-simplex σ is the convex hull of k + 1 linear independent vectors or vertices in R d , with orientation induced by the ordering of the vertices σ = [n 0 , . . . , n k ]. The mass |σ| 0 of a k-simplex σ is its k-dimensional Hausdorff's measure. (In particular the mass of any 0-simplex is 1.) A simplicial k-chain c is a formal (finite) sum of k-simplices with real coefficients. The mass |c| 0 of a simplicial chain c = I i=1 a i σ i is defined as

|c| 0 = I i=1 |a i ||σ i | 0 . We denote by C k (resp. C k (T )) the space of simplicial k-chains in R n (resp. supported in T ).
We denote ∆ k (T ) the set of k-subsimplices of T . If d = 3 then ∆ 0 (T ) is the set of the four vertices of T , ∆ 1 (T ) of its edges, ∆ 2 (T ) of its faces, and ∆ 3 (T ) ≡ T . Note that we will consider also other k-simplices supported in T . Whitney's k-forms of degree one [START_REF] Whitney | Geometric Integration Theory[END_REF] are associated with the k-subsimplices of T . For each n ∈ ∆ 0 (T ) the Whitney's 0-forms w n is the barycentric function λ n associated with n. For each σ = [n 0 , . . . , n k ] ∈ ∆ k (T ) we have w σ = k j=0 (-1) j λ nj dw σ-nj being w σ-nj the Whitney's (k -1)-form associated with the (k -1)-subsimplex of T obtained by eliminating the vertex n j from σ, and d the exterior derivative. In particular w T = |T | -1 0 . Whitney's k-forms are element of the dual basis for the weights on the k-subsimplices of T , namely, σ w σ = δ σ,σ for σ, σ ∈ ∆ k (T ). Note that Whitney's k-forms have proxies with components given by polynomials of degree 1 if 0 ≤ k < d and 0 if k = d.

Whitney's k-forms of degree r > 1 [START_REF] Rapetti | Whitney forms of higher degree[END_REF] are the elements of the space

P - r λ k (T ) = Span{λ α w s : s ∈ ∆ k (T ) and α ∈ I(d, r -1)} being I(d, r -1) = {(α 0 , ..., α d ), α i ∈ N, i α i = r -1} and λ α = λ α0 0 ...λ α d d . We denote C 0 Λ k (R d ) the Banach's space of continuous k-differential forms w with bounded C 0 norm: w C 0 < ∞ and C 0 Λ k (T ) the space of restrictions of C 0 Λ k (R d ) to T .
The norm w C 0 has to be intended as the maximum norm of the proxy of w.

Polynomial interpolation of forms

Given a set X k,r = {s 1 , ..., s N k,r } of N k,r = dim(P - r Λ k (T )) distint k-simplices in T (not necessarily subsimplices of T ), the interpolation problem for a given differential k-form w ∈ C 0 Λ k (T ) consists in finding a Whitney's differential k-form, I r w ∈ P - r Λ k (T ), such that si I r w = si w for all s i ∈ X k,r . As in the case of Lagrange's interpolation, chosen a basis B = {w j } N k,r j=1 for the space P - r Λ k (T ) the interpolation problem is well-defined if the generalized Vandermonde's matrix V X k,r ,B with entries (V X k,r ,B ) i,j = si w j , i, j = 1, . . . , N k,r , is invertible. We say that the set X k,r of N k,r k-simplices supported in T is unisolvent for the space P - r Λ k (R n ) if the interpolation problem is well-defined. With any unisolvent set X k,r , it is possible to associated a dual basis {w

X k,r j } N k,r j=1 of P - r Λ k (R n ) (namely, such that si w X k,r j = δ i,j
). Examples of unisolvent sets of k-simplices for the space P - r Λ k (R n ) are given in [START_REF] Rapetti | Whitney forms of higher degree[END_REF] and [START_REF] Alonso Rodríguez | Minimal sets of unisolvent weights for high order Whitney forms on simplices. Enumath 2019 procs[END_REF].

Definition 2. Given a set X k,r = {s 1 , ..., s N k,r } of N k,r k-simplices supported in the d-simplex T that is unisolvent for the space P - r Λ k (T ), the Lebesgue's function L X k,r (T ) : C k (T ) → R + is defined as Proof: To prove that Λ X0,r ≤ Λ 0 X0,r , we recall that the 0-simplices supported in T are points x of T , |x| 0 = 1, and φ

L k X k,r (c) = N k,r j=1 |s j | 0 c w X k,r j with {w X k,r j } N k,r j=1 ⊂ P - r Λ k (T ) the dual basis associated with X k,r . The Lebesgue's constant is then Λ k X k,r = sup c∈C k (T ) L k X k,r (c)
= I i=1 |a i |. Therefore Λ 0 X0,r = sup c∈C0(T ) L 0 X 0,r (c) |c|0
≤ Λ X0,r .

For any L 1 -differential k-form w we define w 0 := sup σ =0 | σ w| |σ|0 , where the supremum is taken over all k-subsimplices of T . If w ∈ C 0 Λ k (T ) then w 0 = w C 0 . (See, e.g., [START_REF] Harrison | Continuity of the integral as a function of the domain[END_REF].) Proposition 2. Let X k,r (T ) = {s 1 , ..., s N k,r } be a unisolvent set of N k,r ksimplices supported in the d-simplex T that is unisolvent for the space P - r Λ k (T ).

For each differential k-form w ∈ C 0 Λ k (T ) it holds

w -I r w 0 ≤ (1 + Λ X k,r ) w -w * 0 ( 4
)
where w * is the best-fit reconstruction of w for the given norm • 0 , namely w * ∈ P - r Λ k (T ) and it verifies w -w * 0 ≤ w -z 0 for all z ∈ P - r Λ k (T ). Proof: First we notice that I r w * = w * and

w -I r w 0 ≤ w -w * 0 + I r (w * -w) 0 ≤ (1 + I r 0 ) w -w * 0 (5)
being I r 0 := sup w∈C 0 Λ k (T )

Irw 0 w 0 . Recalling that I r w = N k,r j=1 sj w w X k,r j we have c I r w = N k,r j=1 sj w c w X k,r j ≤ N k,r j=1 s j w |sj |0 |s j | 0 c w X j ≤ max 1≤j≤N k,r s j w |sj |0 N k,r j=1 |s j | 0 c w X k,r j ≤ w 0 L k X k,r (c) .
It is easy to check that w 0 = sup c∈C k | c w| |c|0 , so

I r w 0 = sup c∈C k c I r w |c| 0 ≤ w 0 sup c∈C k L k X k,r (c) |c| 0 = w 0 Λ k X k,r
and I r 0 := sup w∈C 0 Λ k (T )

Irw 0 w 0 ≤ Λ k X k,r .
Replacing in [START_REF] Chung | On lattices admitting unique Lagrange interpolations[END_REF] we obtain (4).

Some remarks and conclusions

We have generalized the definition of the Lebesgue's function and constant to the case of k-form interpolation in P - r Λ k , the space of "trimmed" polynomial differential forms widely used in the finite elements exterior calculus (see [START_REF] Arnold | Finite element exterior calculus, homological techniques, and applications[END_REF]). This extension is very natural when using the weights on subsimplices as degrees of freedom due to their clear physical meaning. For r > 1, the subsimplices have to be viewed as an enrichment of the geometrical description of the weights' support, as if we were computing a more accurate approximation of circulations (k=1), fluxes (k=2), etc. As in the classical setting of Lagrange's polynomial interpolation (that in fact correponds to the case k = 0), the Lebesgue's constant is a quality indicator when selecting among unisolvent sets of degrees of freedom, the ones that minimize the local interpolation error in the maximum norm. Applications are postponed to future works.

|c|0 .Proposition 1 .

 |c|01 If k = 0 then Definition 2 coincides with Definition 1.

  hand, if c = I i=1 a i x i ∈ C 0 (T ) then, taking into account that |s i | 0 = 1 for each s i ∈ X 0,r , one has | = |c| 0 Λ Xr , because for a chain c = I i=1 a i x i ∈ C 0 (T ) one has |c| 0

Finantial support from French MathIT program ANR-15-IDEX-01 and Italian PRIN'17 project NA-FROM-PDEs.


Fanny Orlhac

Anne-Capucine Rollet

Irène Buvat

Jacques Darcourt

Véronique Bourg

Christophe Nioche

Charles Bouveyron

Nicholas Ayache

Olivier Humbert

Identifying a reliable radiomic signature from scarce data: illustration for 18F-FDOPA PET images in glioblastoma patients

Materials and Methods: 84 patients with a clinical and MRI suspicion of recurrent glioblastoma were retrospectively included. Each patient underwent a 18F-FDOPA PET-CT scan. For each patient, the suspicious lesion was segmented and 49 radiomic features were calculated using LIFEx [1]. Our goal was to distinguish between tumor recurrence and radiation-induced necrosis as confirmed on pathological data, or on a 3-month clinical/imaging follow-up. A screening procedure was developed to identify a signature using leave-one-out (LOO) cross-validation. The procedure involved: 1) selection of all features with a p-value of univariate Wilcoxon test lower than alpha varying from 0.005 to 0.10 for (N-1) learning patients, 2) based on these features, selection of only one feature among correlated features using a Pearson correlation cut-off R varying from 0.95 to 0.50, 3) building of a radiomic signature involving the resulting features using a linear discriminant analysis, 4) test of the model on the Nth patient, 5) characterization of the model performance using the Youden Index (Y=sensitivity+specificity-1). The final model selection was based on the consistency of Y values as a function of alpha and R, on the consistency of selected features between the different models, and favored models involving a low number of features. To test the reliability of the selected signature, we repeated the process by excluding one patient using a jackknife procedure (ie, 84 LOO of 83 patients each). We compared the results with those obtained with the same alpha and R when randomly assigning a label to each patient (sham task).

Results: 61 patients had tumor recurrence and 23 had radiation necrosis. Visual interpretation yielded Y equal to 0.35 (Lizarraga scale, Se=100%, Sp=35%). Using LOO, 10/28 radiomic models had Y>0.35. The largest Y (Y=0.49, Se=62%, Sp=87%) were obtained with 4 features on average, reflecting the volume, sphericity and heterogeneity (GLCM_Correlation, GLCM_Contrast) of the lesion uptake. Using the jackknife procedure, Y was 0.47±0.06 (range: [0.32;0.56]), significantly higher (Wilcoxon p < 0.05) than for the sham task (Y=0.06±0.18, range: [-0.52;0.47]).

Conclusion:

The proposed systematic screening procedure enabled the identification of a parsimonious radiomic signature from scarce data.

  References: [1] Nioche et al. Cancer Res 2018.


Fanny Orlhac

Thibaut Cassou-Mounat

Jean-Yves Pierga

Marie Luporsi

Christophe Nioche

Charles Bouveyron

Nicholas Ayache

Nina Jehanno

Alain Livartowski

Irene Buvat

Can we identify "twin patients" to predict response to neoadjuvant chemotherapy in breast cancer?

Can we identify "twin patients" to predict response to neoadjuvant chemotherapy in breast cancer?

Fanny Orlhac1, Thibaut Cassou-Mounat1,2, Jean-Yves Pierga3, Marie Luporsi2,1, Christophe Nioche1, Charles Bouveyron4, Nicholas Ayache5, Nina Jehanno2,1, Alain Livartowski6, Irene Buvat1 Objectives: Most supervised learning approaches currently applied in radiomics consist in classifying patients in groups (eg, responder versus non-responder, short overall survival versus long overall survival). With these methods, individual information from each patient is used only to assign a patient to a group. To preserve detailed information of each patient, we are developing an alternative approach that consists in identifying a "twin patient" based on radiomic features and clinical parameters in a patient database, i.e. another patient with feature values similar to the ones observed in the tested patient. Here, we studied whether this approach could predict the response to neoadjuvant chemotherapy in breast cancer patients.

Methods: 117 patients with a triple-negative breast cancer were included in this study. All patients underwent a baseline 18F-FDG PET/CT using Gemini GXL 16 (Philips, 41 patients) or Discovery 710 Elite (GE, 76 patients) with a standard imaging protocol, before a neo-adjuvant chemotherapy associating anthracycline and taxane. The pathological response was assessed on the surgical specimen after chemotherapy according to the Residual Cancer Burden (RCB score). Based on PET images, the primary lesion of each patient was segmented using a threshold set to 40% of SUVmax, and a 6 mm thick ring around the tumor region was also used to measure the peri-tumoral metabolic activity. In each resulting volume of interest, we computed 48 radiomic features using LIFEx software (intensity resampling: 128 gray-levels between 0 and 40 SUV; spatial resampling: 2x2x2 mm) [1]. Radiomic features were harmonized between the two PET devices using ComBat [2]. The profile of each patient included 96 radiomic features and 3 clinical variables (age, Ki-67 expression, body mass index), each expressed as z-score. Using a leave-one-out approach, we computed the Euclidian distance between the profile of the tested patient and the profiles of all patients of the database. The smallest distance was used to identify the twin of the tested patient. This procedure was repeated by defining profiles including only 2, 4 and 6 features using a systematic feature selection process. The performance was evaluated using the Youden Index (YI=Sensitity+Specificity-1) to predict the response to neoadjuvant chemotherapy.

Results:

In our cohort, 68 patients were identified as responders to therapy and 49 patients as non-responders. Without feature selection, YI was 0.13 (Se=45% ; Sp=68%). When using 2 features only, the best performance was obtained for the combination of Entropy_Tumor and LZE_Ring with YI=0.36 (Se=59% ; Sp=76%). With 4 features, the best YI (=0.50) was obtained by adding Energy_Ring and SRE_Ring to Entropy_Tumor and LZE_Ring (Se=73% ; Sp=76%). With 6 features, the best combination was obtained by adding LZE_Tumor and LZHGE_Ring, yielding a YI of 0.56 (Se=80% ; Sp=76%). When we selected features, clinical variables were never included in the best combination, and radiomic features extracted from the peri-tumoral regions (ring) improved the identification of relevant twins.

Conclusion:

We demonstrated that the so called "twin patients" defined as sharing a number of similar radiomic features were likely to present similar response to neo-adjuvant chemotherapy. Feature selection improved the performance by including only features relevant to the prediction task into the profile. Unlike the conventional group-based patient management, the twin approach uses the detailed individual information of each patient previously treated. Further studies involving an independent cohort and more features in the combination are underway to further explore the potential of this concept. The identification of patient similarities also acts as a discovery approach to formulate new biological hypotheses, for instance, highlighting here the association between peri-tumoral heterogeneity and response to therapy.

1.

  Laboratoire d'Imagerie Translationnelle en Oncologie (LITO) Université Paris-Saclay/Inserm/Institut Curie Orsay France 2. Department of Nuclear Medicine Institut Curie Paris France 3. Department of Medical Oncology Institut Curie Paris France 4. Maasai team Université Côte d'Azur, Inria Sophia-Antipolis France 5. Epione team Université Côte d'Azur, Inria Sophia-Antipolis France 6. Direction des Data Institut Curie Paris France


Yoseline Rosales-Cabara

Giovanni Manfredi
email: giovanni.manfredi@ipcms.unistra.fr

Gabriel Schnoering

Paul-Antoine Hervieux
email: ‡hervieux@ipcms.unistra.fr

Laurent Mertz
email: laurent.mertz@nyu.edu

Cyriaque Genet
email: genet@unistra.fr

Rapid Communications

We propose an optimization strategy to control the dynamics of a stochastic system transferred from one thermal equilibrium to another and apply it experimentally to a Brownian particle in an optical trap under compression. Based on a variational principle that treats the transfer duration and the expended work on an equal footing, our strategy leads to a family of protocols that is either optimally cheap for a given duration or optimally fast for a given energetic cost. This approach unveils a universal relation t W ( t W ) opt between the transfer duration and the expended work. We verify experimentally that the lower bound is reached only with the optimized protocols.

protocols that are optimized from both points of view. The possibility for optimal control turns out to be particularly relevant in the field of stochastic engines, where it is necessarily related to the global figure of merit of the system [START_REF] Blickle | Realization of a micrometer-sized stochastic heat engine[END_REF][START_REF] Martínez | Brownian carnot engine[END_REF].

In order to derive such optimal protocols, we adopt an approach that treats both the duration of the transfer and the expended work in a completely symmetric way. Our strategy is implemented on an optical trap that determines a simple framework (harmonic potentials) where theory can be elaborated in perfect tandem with experiments. This provides a capacity for exploring optimization with two ingredients. First, each protocol is defined by a path in the phase space (κ, s), where κ is the stiffness of the optical trap and s is the variance of the position of the particle. Second, we construct a functional J[κ, s] that is composed of two terms, corresponding, respectively, to the total work and total duration, with a Lagrange multiplier λ regulating the trade-off between these two quantities. Minimizing the above functional with fixed boundary conditions yields the desired optimal protocol κ λ (s). For instance, λ 1 gives a protocol of low energetic cost but long duration, while λ 1 gives a fast protocol requiring a large amount of work.

Remarkably, this treatment leads to a universal relation t W

( t W ) opt between the transfer duration t and the expended work W (in excess of the equilibrium freeenergy difference), where the lower bound depends exclusively on the initial and final states and is reached only under optimal control conditions. This result unveils a fundamental feature that underpins all optimization procedures in stochastic thermodynamics [START_REF] Sekimoto | Stochastic Energetics[END_REF][START_REF] Ciliberto | Experiments in Stochastic Thermodynamics: Short History and Perspectives[END_REF]. In particular, we demonstrate that protocols optimized under the further constraint of a limited speed of variation for κ (s) do converge towards work-duration optimal ones.

We optically trap a Brownian particle-a polystyrene microsphere of radius 500 nm-in water at room temperature inside a harmonic potential; see the Supplemental Material, Sec. A [15], and [START_REF] Schnoering | Three-Dimensional Enantiomeric Recognition of Optically Trapped Single Chiral Nanoparticles[END_REF][START_REF] Schnoering | Thermally Limited Force Microscopy on Optically Trapped Single Metallic Nanoparticles[END_REF]. We record the instantaneous motion x(t ) of the microsphere along the trap optical axis. With the trap stiffness κ proportional to the trapping laser intensity I, it is possible to define, by modulating I (t ), a given protocol κ (t ) for the transfer from an initial thermal equilibrium at time t i to a final equilibrium at time t f . By performing a series of N identical protocols on the trapped microsphere, we build a statistical ensemble of trajectories that yields a probability density function (PDF) of positions x. The dynamics of the system is described by the variance s(t ) extracted from the PDF of positions, which evolves as [START_REF] Manoliu | Correlation in the Langevin theory of Brownian motion[END_REF][START_REF] Gardiner | Stochastic Methods[END_REF]

γ ds dt = -2κ (t )s + 2Dγ , ( 1
)
where γ is the Stokes drag coefficient and D = k B T /γ ∼ 0.4 μm 2 /s is the Brownian diffusion coefficient fixed by the temperature T (room temperature) of water and the Boltzmann constant k B . Equation ( 1) fully determines the statistical properties of the system where the initial and final equilibria correspond to the stationary solutions s i κ i = s f κ f = k B T with the PDF that remain Gaussian for all intermediate times between t i and t f . The cumulative energetics involved during the protocol is directly related to the time evolution of s(t ), giving the ensemble average expended work [START_REF] Ciliberto | Experiments in Stochastic Thermodynamics: Short History and Perspectives[END_REF][START_REF] Sekimoto | Langevin equation and thermodynamics[END_REF]. Our purpose is to control the dynamics of the microsphere so that the transfer between the two equilibria is optimal with respect to both duration and energetics. If one instantaneously switches the trap stiffness from κ i to κ f > κ i (i.e., closing the trap in a steplike way), the typical relaxation time to the new equilibrium is given by τ relax = 2γ /κ f . Our optimization strategy starts with the idea of using the variance s as the independent variable of the problem, instead of the time t. This is possible whenever the function s(t ) is monotonic and it enables us to express the control parameter as κ (t ) = κ[s(t )]. The advantage of this approach is that we can easily write down, as functionals of κ (s), both the transfer duration,

W (t ) = 1 2 t t i dts(t ) κ (t ) and dissipated heat Q(t ) = -1 2 t t i dt ṡ(t )κ (t )
t[κ (s)] ≡ t f -t i = 1 2 s f s i γ ds Dγ -s κ (s) , (2)
and the expended work,

W [κ (s)] = - 1 2 s f s i κ (s)ds + 1 2 (κ f s f -κ i s i ), (3)
where the second term on the right-hand side vanishes because the initial and final configurations are thermally equilibrated. Next, we define the functional to be minimized as twice the sum of W and t,

J[κ (s)] = s f s i γ ds Dγ -s κ (s) -λ s f s i κ (s)ds, ( 4
)
where λ is a Lagrange multiplier that regulates the trade-off between transfer duration and work. Within this framework, the optimization strategy can be interpreted as the search for the trajectory in the (s, κ ) space that minimizes J[κ (s)] while keeping the extrema fixed at equilibrium, i.e.,

s i κ i = s f κ f = k B T = Dγ . Once written as J = s f s i L[s, κ (s)
]ds, this functional can be minimized using the standard Euler-Lagrange equation d/ds(∂L/∂ κ ) -∂L/∂ κ = 0, where κ ≡ d κ/ds, yielding the following solution:

s κ (s) = Dγ + γ s/λ, (5)
for a protocol that eventually closes the trap with κ f > κ i [21]. Equation ( 5) encapsulates the main result obtained so far. For instance, the quasistatic solution (ṡ ≈ 0) is obtained by taking λ → ∞, which yields an infinite duration but the smallest possible expended work, W QS = 1 2 Dγ ln(κ f /κ i ), equal to the free-energy difference, as expected for a quasistatic process. For finite λ and making use of Eq. ( 5), Eq. ( 1) can be rewritten as ṡ = -2 √ s/γ λ, which possesses the general solution

s(t ) = ( √ s i -t/ √ γ λ) 2 .
Inserting this expression into Eq. ( 5) yields the optimal evolution of the trap stiffness,

κ (t ) = Dγ + √ γ s i /λ -t/λ ( √ s i -t/ √ γ λ) 2 , ( 6
)
which defines our protocol for the optimized transfer.

It is important to stress that in our case, the Euler-Lagrange equation is purely algebraic so that one cannot enforce the initial and final conditions. Thus, except for the quasistatic limit, Eq. ( 6) does not satisfy the conditions for which the system is at thermal equilibrium in the initial and final states. In order to ensure that s i, f κ i, f = Dγ , we need to add to Eq. ( 6) two discontinuities (as already noticed in [START_REF] Schmiedl | Optimal Finite-Time Processes in Stochastic Thermodynamics[END_REF][START_REF] Schmiedl | Efficiency at maximum power: An analytically solvable model for stochastic heat engines[END_REF]). The optimal protocol thus consists of three successive sequences:

(1) At time t = t i , the trap stiffness is suddenly changed

from κ i = Dγ /s i (initial equilibrium) to κ (t + i ) = κ + i , such that κ + i -κ i = √ γ /(λs i )
, while keeping the variance equal to s i .

(2) Between t + i and t - f , the stiffness varies according to Eq. ( 6), reaching κ (t - f ) ≡ κ - f = Dγ /s f + γ /(λs f ). (3) At time t = t f , the stiffness is suddenly changed from κ - f to κ f = Dγ /s f (final equilibrium), while keeping the variance equal to s f . Experimentally, we have performed, on the trapped microsphere, a series of N = 2 × 10 4 such optimal protocols, forcing the microsphere to relax to thermal equilibrium within a time t chosen to be shorter than τ relax . Figure 1 displays s(t ) between two equilibria for a shortened duration t ∼ τ relax /10, where τ relax = 2γ /κ f = 3.22 ± 0.09 ms. The reduction in s(t ) from its initial value corresponds to the fact that the trap is stiffer at the end of the protocol with κ f /κ i ∼ 1.85. It is in excellent agreement with the theory.

This reduction in the transfer duration has an energetic cost that can be evaluated for each sequence and measured experimentally by evaluating the cumulative work W (t ) from the recorded evolution of κ (s). Figure 2 shows the time evolution of W (t ) split into three sequences. First, the quantity of work, W (1) = s i (κ + i -κ i )/2, is injected instantaneously into the system at the time t i as the trap is suddenly stiffened from κ i to κ + i . During the second sequence, the injection of work continues as the trapping volume is progressively reduced, reaching W (2)

= W QS + √ γ /λ[(s 3/2 i -s 3/2 f )/3 - (s 1/2 i -s 1/2 f )/2] at t = t - f , when κ = κ - f .
Finally, the trap is suddenly expanded at t = t f , and the system instantly reaches its final equilibrium state, delivering to the thermal bath a quantity of work equal to W (3) = s f (κ f -κ - f )/2. For t > t f , the thermal steady state is characterized by W = Q. ). Inset: The experimental optimal protocol κ (t ), normalized to the initial stiffness κ i .

In contrast, heat is continuously dissipated from the microsphere to the thermal bath, as seen in Fig. 2, from the monotonic increase of the ensemble average cumulative heat Q(t ) throughout the protocol. This evolution is almost exactly linear in time, which corresponds to a constant production of entropy. We stress again that the experimentally measured values of the heat and work injected in and extracted from the system are in excellent agreement with the theoretical predictions.

The total work expended throughout the optimal protocol is evaluated by adding the contributions from each of the three sequences described above. One obtains

W opt = W QS + γ /λ s 1/2 i -s 1/2 f . ( 7
)
Similarly, the total duration of the optimal protocol is obtained by inserting Eq. ( 5) into Eq. ( 2), yielding

t opt = γ λ s 1/2 i -s 1/2 f . ( 8
)
The above expressions clearly show that our optimization procedure is perfectly symmetric as far as duration and work are concerned and that the trade-off between these two quantities is governed by the Lagrange multiplier λ. Indeed, one can choose λ using Eq. ( 8) to fix the total duration and then the minimum expended work will be given by Eq. ( 7); or, alternatively, one can determine λ through Eq. ( 7) to fix the total work and then the minimum duration of the process will be given by Eq. ( 8). This leads us to define the "excess work" of the optimal protocol as W opt ≡ W opt -W QS and to note that the product

t opt W opt = γ /2(s 1/2 i -s 1/2
f ) 2 is independent of λ and only depends on the initial and final states [23]. This equality fixes the mutually exclusive relation between transfer duration and expended work under optimal control. It corresponds to the frontier value of a universal exclusion region, t W γ /2( √ s i -√ s f ) 2 , that bounds from below all protocols that are not optimal.

The frontier can be drawn experimentally by measuring W opt for a series of optimal protocols with different transfer durations t. By normalizing each measured value of W opt to the associated value of γ /2( √ s i -√ s f ) 2 , one can test the universal nature of the bound. This is clearly confirmed in Fig. 3, with all the optimal solutions implemented experimentally falling precisely on the 1/ t curve. To further prove that the frontier corresponds to a lower bound, we have verified experimentally that the { t, W } coordinates of typical nonoptimal protocols all fall above the expected bound, as displayed in Fig. 3. A salient feature of our optimal control procedure is represented by the sudden jumps in stiffness that have to augment the solution of Eq. ( 6) in order to comply with thermally equilibrated initial and final configurations. From an experimental point of view, such discontinuities do not constitute a weakness of the procedure, as they correspond to finite and measurable quantities of work exchanged between the bath and the system [START_REF] Plata | Optimal work in a harmonic trap with bounded stiffness[END_REF][START_REF] Aurell | Boundary layers in stochastic thermodynamics[END_REF]. But it is interesting to stress that one asset of our variational strategy is its capacity to construct smooth protocols that are as close as desired to the optimal ones. For this, we need to control the derivatives of the function κ (s) by adding the gradient term 4), with a second Lagrange multiplier ε that acts as a regularization parameter by removing FIG. 3. Extracted excess works W opt revealing the mutually exclusive relation between W opt and t for a series of protocols. Optimal protocols (squares) for transfer durations t = τ relax /n, with successively n ∼ 34, 30, 22, 16, 10, 6, 3, and 2, "engineered swift equilibration" protocol [START_REF] Martínez | Engineered swift equilibration of a Brownian particle[END_REF] (black diamond) defined for t = 3.47 × 10 -4 s ∼ τ relax /10 (also in inset), steplike protocol (black star) at τ relax ∼ 3.22; see details in the Supplemental Material, Sec. B [15]. All optimized coordinates { t, W opt /[γ /2( √ s i -√ s f ) 2 ]} precisely fall (within error bars) on the 1/ t curve, whereas the values corresponding to the nonoptimal protocol fall above the universal bound. Insets: coordinates measured for smooth (thus suboptimal) protocols for n ∼ 22 and n ∼ 10, and smoothness parameters ε = 5 × 10 -6 , ε = 10 -6 , and ε = 0, expressed in units of s 2 i /(Dκ 2 i ). Such smooth protocols are defined using the same Lagrange multiplier λ as their associated optimal protocols. For each case, the product t W converges to the optimal lower bound (solid blue line) as ε → 0.

s f s i | κ (s)| 2 ds to the functional J[κ (s)] in Eq. (
the discontinuities in κ (s). Hence, we arrive at the modified Euler-Lagrange equation 2ε d 2 κ/ds 2 = γ s/(Dγs κ ) 2 -λ, which can be solved numerically as a boundary value problem, with initial and final conditions at thermal equilibrium, κ (s i, f ) = Dγ /s i, f . Once the solution κ (s) is known, the time evolution of the variance s(t ) is found by integrating Eq. ( 1) (more details are given in the Supplemental Material, Sec. C [15]).

Using the same values of λ that defined the optimal protocols with, respectively, t ∼ τ relax /22 and t ∼ τ relax /10 (Fig. 3, inset), we implemented two smooth protocols for two different values of the Lagrange multiplier. As shown in Fig. 4, the smooth protocols closely follow the optimal ones, except near the beginning and the end of the process, where they approach the equilibrium states in a continuous way. For the same value of λ, smooth protocols give slightly longer transfer durations than the optimal protocol but, as expected, the expended work is slightly smaller in the smooth case than in the optimized limit. The nonoptimal character of the smooth protocols is clearly seen in the insets of Fig. 3, where all { t, W } coordinates lie above the universal bound, and only converge towards it in the ε → 0 limit.

In conclusion, we have devised a family of optimal protocols that transfer an optically trapped microsphere between FIG. 4. Comparison of the cumulative energetics-expended work W (t ) (blue curves) and dissipated heat Q(t ) (red curves)between an optimal protocol defined for t = 1.72 × 10 -4 s ∼ τ relax /22 and two smooth protocols with ε = 5 × 10 -6 and ε = 10 -6 , expressed in units of s 2 i /(Dκ 2 i ), and identical value of λ = (2.97 ± 0.12) × 10 16 s/J. As seen in the insets of Fig. 3, although the smooth protocols involve slightly less expended work (1.36 ± 0.06 k B T for ε = 5 × 10 -6 and 1.65 ± 0.06 k B T for ε = 10 -6 ) than the optimal one (1.69 ± 0.06 k B T ), they correspond to longer transfer durations (2.48 × 10 -4 s for ε = 5 × 10 -6 and 2.14 × 10 -4 s for ε = 10 -6 ). Inset: Superimposed evolutions κ (t ) vs s(t ) for the three protocols, showing the continuous nature of the smooth protocol and illustrating the progressive convergence to the optimal protocol in the ε → 0 limit. For each protocol, the curves are normalized to the corresponding κ i for κ (t ) and s f for s(t ). two equilibria, minimizing both the transfer duration and the associated energetic cost. Within such protocols, the tradeoff between duration and work can be modulated at will by tuning a single Lagrange multiplier given by our variational approach. A key result of our work is to show that the product t W is bounded from below, in a way reminiscent of energy-time uncertainty relations. Similar bounds were noticed in earlier works [START_REF] Sekimoto | Stochastic Energetics[END_REF][START_REF] Ciliberto | Experiments in Stochastic Thermodynamics: Short History and Perspectives[END_REF], but only for some special cases. Here, our bound is universal (it depends exclusively on the initial and final states) and is only reached for the optimal protocol. Further extending the present results to quantum systems may open new interesting perspectives in the burgeoning field of quantum stochastic thermodynamics [START_REF] Elouard | The role of quantum measurement in stochastic thermodynamics[END_REF][START_REF] Vinjanampathy | Quantum thermodynamics[END_REF][START_REF] Roßnagel | A single-atom heat engine[END_REF][START_REF] Cavina | Optimal thermodynamic control in open quantum systems[END_REF].

FIG. 1 .

 1 FIG.1. Time evolution of the variance s(t ) for an optimal protocol of duration of t = 3.47 × 10 -4 s ∼ τ relax /10, indicated by the vertical solid line. Starting at time t i at thermal equilibrium with κ i = 2.77 ± 0.08 pN/μm and a mean s i = 1.48 × 10 -15 m 2 , the variance is extracted from the PDF of positions and normalized to the final equilibrium state associated with the plateau-averaged value s f = 7.75 × 10 -16 m 2 reached after t f , corresponding to κ f = 5.22 ± 0.15 pN/μm. The Lagrange multiplier λ = (1.27 ± 0.02) × 10 17 s/J associated with this protocol is determined by the set of values ( t, s i , s f ) and Eq. (8). The superimposed black continuous line is the theoretical time evolution of the optimized variance,s(t ) = ( √ s it/ √ γ λ)2 , calculated with the measured values. The experimental error bars correspond to a 95% confidence interval for s(t ), including calibration uncertainties (see more details in the Supplemental Material, Sec. A [15]). Inset: The experimental optimal protocol κ (t ), normalized to the initial stiffness κ i .

FIG. 2 .

 2 FIG. 2. Energetics associated with the optimal protocol described in Fig. 1 for t ∼ τ relax /10. The ensemble average cumulative work W (t ) (blue curve) and heat Q(t ) (red curve) are measured with respect to the initial thermal equilibrium. The experimental error bars are determined by the propagation of variance and calibration (i.e., stiffness) uncertainties (see the Supplemental Material, Sec. A [15]). At t > t f , the system has reached a thermal steady state with W (t > t f ) = (0.981 ± 0.059) k B T ∼ Q(t > t f ) = (0.983 ± 0.060) k B T . Inset: The control parameter κ (s), with the arrows corresponding to the time evolution.

  This work was supported in part by Agence Nationale de la Recherche (ANR), France, ANR Equipex Union (Grant No. ANR-10-EQPX-52-01), the Labex NIE projects (Grant No. ANR-11-LABX-0058-NIE), and USIAS within the Investissements d'Avenir program (Grant No. ANR-10-IDEX-0002-02). Y.R.-C. is a member of the International Doctoral Program of the Initiative d'Excellence of the University of Strasbourg, whose support is acknowledged. L.M. is supported by the National Natural Science Foundation of China, Research Fund for International Young Scientists under the Project No. 1161101053 and the Young Scientist Program under the Project No. 11601335.


Fabienne Quilès

Danielle Barth

Oliver Peric

Georg E Fantner

Grégory Francius
email: gregory.francius@univ-lorraine.fr

Parietal structures of Escherichia coli can impact the D-cateslytin antibacterial activity

Keywords: parietal structures, antimicrobial peptide, D-cateslytin, mechanism of action, AFM, epifluorescence microscopy, infrared spectroscopy, Escherichia coli

Bacterial resistance to conventional antibiotics is of major concern. Antimicrobial peptides (AMPs) are considered as excellent alternatives. Among them, D-cateslytin (D-Ctl, derivative of a host defense peptide), has shown high efficiency against a broad spectrum of bacteria.

The first target of AMPs is the outer membrane of the bacterium. However, the role of bacterial cell-wall structures on D-Ctl mechanism of action has not yet been understood. In this study, we investigated the activity of D-Ctl on two isogenic strains of E. coli: one is nude of any parietal structures, the other constitutively overexpresses only Type 1 fimbriae. We studied the damages caused by D-Ctl at several initial concentrations of bacteria and D-Ctl, and times of exposure to D-Ctl were examined using a combination of epifluorescence microscopy, atomic force microscopy (AFM), and Fourier transform infrared spectroscopy in attenuated total reflectance mode (ATR-FTIR). The analysis of nanomechanical and spectrochemical properties related to the antibacterial mechanism showed a concentration dependent activity. Whereas the membrane permeabilization was evidenced for all concentrations of D-Ctl and both mutants, no pore formation was observed. The bacterial stiffness is modified dramatically concomitantly to major membrane damages and changes in the spectral fingerprints of the bacteria. In case of the occurrence of Type 1 fimbriae only, an intracellular activity was additionally detected. Our results evidenced that D-Ctl activity is highly impacted by the cell-wall external structures and surface properties of the bacteria.

Introduction

Bacterial resistance to conventional antibiotics has been increasing at an alarming rate during the last two decades, affecting dramatically our ability to treat common infections. [START_REF] Williams | Challenges in developing new antibacterial drugs[END_REF] In addition, insufficient investment in antimicrobial therapies resulted in a lack of the discovery of new families of active molecules. [START_REF] Williams | Challenges in developing new antibacterial drugs[END_REF][START_REF] Da Cunha | Antibiotic Discovery: Where Have We Come from, Where Do We Go?[END_REF][START_REF] Silver | Challenges of antibacterial discovery[END_REF] This has led the World Health Organization (WHO) to raise the alarm against antimicrobial resistance and especially the resurgence of multi-drug resistant bacteria. It is particularly true for Gram-negative bacteria. [START_REF] Silver | Challenges of antibacterial discovery[END_REF][START_REF] Tacconelli | Discovery, research, and development of new antibiotics: the WHO priority list of antibiotic-resistant bacteria and tuberculosis[END_REF] The urgent need to develop new molecules as alternatives to conventional antibiotics has led to consider to use peptides with antimicrobial properties.

Antimicrobial peptides (AMPs) have a minimum inhibitory concentration (MIC) often in the range of a few µM. Despite a strong antimicrobial activity at low concentrations, their mode of action in killing bacteria or in inhibiting bacterial growth remains not fully understood. [START_REF] Corrêa | Fundamentals on the molecular mechanism of action of antimicrobial peptides[END_REF][START_REF] Kumar | Antimicrobial Peptides: Diversity, Mechanism of Action and Strategies to Improve the Activity and Biocompatibility In Vivo[END_REF] Most AMPs are believed to interact with the membranes where they induce permeabilization.

The subsequent lysis or damage to bacteria is then observed. [START_REF] Shai | Mode of action of membrane active antimicrobial peptides[END_REF][START_REF] Wenzel | Small cationic antimicrobial peptides delocalize peripheral membrane proteins[END_REF] It is generally accepted that AMPs can form transmembrane channels by self-aggregation or multimerization, [START_REF] Jean-Francois | Aggregation of cateslytin beta-sheets on negatively charged lipids promotes rigid membrane domains. A new mode of action for antimicrobial peptides?[END_REF][START_REF] Van Der Weide | Investigations into the killing activity of an antimicrobial peptide active against extensively antibiotic-resistant K. pneumon iae and P. aeruginosa[END_REF] leading to membrane permeabilization and bacterial death. [START_REF] Shai | Mode of action of membrane active antimicrobial peptides[END_REF] The AMP activity on the membrane can be influenced by the parietal structures of the bacteria such as lipopolysaccharides or lipoteichoic acids and the presence of pili and other appendages. [START_REF] Malanovic | Gram-positive bacterial cell envelopes: The impact on the activity of antimicrobial peptides[END_REF][START_REF] Joo | Bacterial strategies of resistance to antimicrobial peptides[END_REF] Indeed, previous works highlighted that external cell wall structures of bacteria act as permeability barriers allowing selective passage of nutrients and exclusion of harmful substances (e.g. antimicrobial agents). [START_REF] Silhavy | The bacterial cell envelope[END_REF][START_REF] Miller | Antibiotic Resistance and Regulation of the Gram-Negative Bacterial Outer Membrane Barrier by Host Innate Immune Molecules[END_REF] Also some non-lytic intracellular modes of action have been reported. [START_REF] Le | Intracellular Targeting Mechanisms by Antimicrobial Peptides[END_REF][START_REF] Scocchi | Non-Membrane Permeabilizing Modes of Action of Antimicrobial Peptides on Bacteria[END_REF] Some antimicrobial peptides can modify and/or inhibit bacterial metabolism, i.e. nucleic acids and protein biosynthesis, cell division, cell wall and lipopolysaccharide biosynthesis. [START_REF] Le | Intracellular Targeting Mechanisms by Antimicrobial Peptides[END_REF][START_REF] Brogden | Antimicrobial peptides: pore formers or metabolic inhibitors in bacteria?[END_REF][START_REF] Jackson | The intracellular target of the antimicrobial peptide thanatin[END_REF] Among the AMPs, host defense peptides (HDPs) are emerging as excellent candidates for the development of new antibiotic strategies. In addition to their antimicrobial action, they exhibit less side effects than other AMPs. [START_REF] Nijnik | Host defence peptides: antimicrobial and immunomodulatory activity and potential applications for tackling antibiotic-resistant infections[END_REF][START_REF] Hancock | The immunology of host defence peptides: beyond antimicrobial activity[END_REF][START_REF] Gupta | Host defense peptides: An insight into the antimicrobial world[END_REF] They are able to kill a broad spectrum of microorganisms with a very low cytotoxicity to mammalian cells. Owing to a general nonspecific action, they have a low level of induced resistance. [START_REF] Joo | Bacterial strategies of resistance to antimicrobial peptides[END_REF][START_REF] Gupta | Host defense peptides: An insight into the antimicrobial world[END_REF] Moreover, some of them have a very interesting therapeutic potential because they can activate the immune system while inhibiting the inflammatory response. [START_REF] Hemshekhar | Functions of Cationic Host Defense Peptides in Immunity[END_REF][START_REF] Schuerholz | Antimicrobial peptides and their potential application in inflammation and sepsis[END_REF] Peptides derived from the proteolysis of chromogranin A are of particular therapeutic interest, and particularly cateslytin. [START_REF] Zaet | D-Cateslytin, a new antimicrobial peptide with therapeutic potential[END_REF][START_REF] Aslam | Cateslytin, a chromogranin A derived peptide is active against Staphylococcus aureus and resistant to degradation by its proteases[END_REF] This fragment of chromogranin A is a linear HDP of 15 residues, and therefore very easy to synthesize for a minimal cost. [START_REF] Zaet | D-Cateslytin, a new antimicrobial peptide with therapeutic potential[END_REF][START_REF] Aslam | Cateslytin, a chromogranin A derived peptide is active against Staphylococcus aureus and resistant to degradation by its proteases[END_REF] Previous studies demonstrated the high efficiency of Dcateslytin (D-Ctl) against a broad spectrum of bacteria including E. coli. [START_REF] Zaet | D-Cateslytin, a new antimicrobial peptide with therapeutic potential[END_REF] It is accepted that the first target of the AMP is the outer membrane of the bacterium. However, no particular attention was paid to the role of bacterial cell-wall structures on D-Ctl efficiency or possible modification of its mode of action from one bacterial strain to another one.

In the present study, we investigated the action of D-Ctl on two strains of E. coli: one is nude of any parietal structures, the other overexpresses only Type 1 fimbriae. [START_REF] Francius | Bacterial surface appendages strongly impact nanomechanical and electrokinetic properties of Escherichia coli cells subjected to osmotic stress[END_REF] Type 1 fimbriae are uniformly distributed on the bacterial surface, and they are known for their surface adherence on epithelial cells. The bacterial attachment allowed by these Type 1 fimbriae is a major factor in bacterial pathogenicity. [START_REF] Antao | Adhesive threads of extraintestinal pathogenic Escherichia coli[END_REF][START_REF] Krogfelt | Bacterial adhesion: genetics, biogenesis, and role in pathogenesis of fimbrial adhesins of Escherichia coli[END_REF][START_REF] Connell | Type 1 fimbrial expression enhances Escherichia coli virulence for the urinary tract[END_REF] The action and the possible influence of these parietal structures were analyzed with a physico-chemical point of view through morphological, nanomechanical, and spectroscopic analyses. This work was performed with a combination of epifluorescence microscopy, atomic force microscopy (AFM), and Fourier transform infrared spectroscopy in Attenuated total reflectance mode (ATR-FTIR). [START_REF] Freudenthal | Discrepancies between Cyclic and Linear Antimicrobial Peptide Actions on the Spectrochemical and Nanomechanical Fingerprints of a Young Biofilm[END_REF][START_REF] Quiles | In situ and real time investigation of the evolution of a Pseudomonas fluorescens nascent biofilm in the presence of an antimicrobial peptide[END_REF] These techniques were combined to monitor in situ the mechanism of action and the damages caused by D-Ctl at several initial concentrations of bacteria and times of exposure. Indeed, beyond the determination of MIC that gives information on the inhibition of growth, it is important to investigate what can happen when a high amount of bacteria is initially present. ATR-FTIR allows the investigation of changes in biochemical compounds of E coli, whereas AFM was used to visualize the local effects of the antimicrobial peptide upon cell morphology and mechanical properties. In addition, the bacterial membrane integrity was evaluated by assays with the BacLight™ kit. Our investigations allow correlation between D-Ctl activity on both nanomechanical properties and bacterial metabolism depending on the antimicrobial agent concentration and the presence of cell-wall external structures.

Materials and Methods

Chemicals and synthetic peptide

Ampicillin, kanamycin and phosphate saline buffer (PBS) were purchased from Sigma-Aldrich, France. D-cateslytin is a positively charged (5+) arginine-rich peptide with the following amino acid sequence: RSMRLSFRARGYGFRGPG. It was purchased from ProteoGenix (Schiltigheim, France, purity > 95%). The peptides were stored at -20°C as a stock solution at 1 g/L in non-pyrogenic sterile water (Aqua B-Braun, Melsungen, Germany).

Structures and physicochemical properties of cateslytin are reported in Figure S1 (see Supporting Information).

Bacterial strain and culture conditions

The bacterial models used in this study are Gram-negative Escherichia coli mutants called E2152 and E2146 kindly provided by Institut Pasteur from Paris. [START_REF] Francius | Bacterial surface appendages strongly impact nanomechanical and electrokinetic properties of Escherichia coli cells subjected to osmotic stress[END_REF] These isogenic strains were constructed from Escherichia coli MG1655. These mutants are non-flagellated and are resistant to specific antibiotics (ampicillin, kanamycin, chloramphenicol and zeocin). Strain E2152 is devoted from the main E. coli parietal structures while strain E2146 constitutively expresses only the external Type 1 fimbriae. E2152 strain was selected as a reference. [START_REF] Francius | Bacterial surface appendages strongly impact nanomechanical and electrokinetic properties of Escherichia coli cells subjected to osmotic stress[END_REF] Bacterial stocks were maintained at -80°C. Bacteria were grown in Lysogeny Broth (LB, Miller, Fluka) at 25 g/L in deionized water (Purelab Option, ELGA). All cultures were grown in a water bath at 37 ± 1°C, and under continuous agitation at 160 rpm. After an overnight sub-culture (16 hours, with antibiotics, i.e. ampicillin and kanamycin), bacteria were cultivated in 25 mL of LB medium (without antibiotics) in 150 mL Erlenmeyer flasks with an initial optical density at 600 nm of 0.05 ± 0.01. The bacteria were taken at the end of the exponential phase when OD 600 reaches the value of 0.50 ± 0.02.

Growth curves and minimum inhibitory concentration (MIC) estimation

The growth curves were obtained by measuring the optical density at 600 nm (OD 600 ) with a Tecan Infinite M200 spectrometer during 24 hours on bacterial suspensions (volume 200 µL) put in a 96-well plate (Nunc) in the presence or not of D-Ctl in LB medium. The bacterial suspension was used either at OD 600 value of 0.50 or diluted to values at 0.10 and 0.001 corresponding to 3.18 ± 0.27×10 8 , 1.05 ± 0.18×10 8 and 2.64 ± 0.33×10 6 CFU/mL, respectively. The measurements were performed at least in triplicate. The MIC of strain E2152 was estimated in LB medium by the two-fold serial dilution assay as described elsewhere. [START_REF] Zaet | D-Cateslytin, a new antimicrobial peptide with therapeutic potential[END_REF] Briefly, the bacterial suspension was diluted to OD 600 = 0.001 in a sterile 96-well plate (Nunc, suspension volume 200 µL). After 24 h of incubation at 37°C without or with D-Ctl at different concentrations, the bacterial growth was assessed by the measurement of OD 600 using a Tecan Infinite M200.

Suspensions for epifluorescence microscopy, infrared spectroscopy and AFM

For epifluorescence and infrared spectroscopy analyses, the antimicrobial assays against planktonic E. coli were performed in duplicate in sterile 96-well plates (Nunc) in a final volume of 200 mL. When the optical density of the bacterial culture reached an OD 600 value of 0.50 ± 0.02, the suspension was diluted in LB to give an OD 600 = 0.10 ± 0.01. The necessary volumes of the stock solution of the peptide at 1 g/L was spotted in the bacterial suspension to a final volume of 200 µL. Sterility and growth controls were sterile LB, and a bacterial suspension without peptide, respectively. The plate was incubated for 1, 4 and 14 hours at 23 ± 1°C in an air conditioned room.

For AFM in-situ experiments, the required volume of D-Ctl mother solution at 537 µM (1 g/L) was added to 1 mL of bacterial suspensions with an OD 600 of 0.50 ± 0.02 to reach final concentrations of 0, 8, 40 and 80 µg/mL. Then suspensions were deposited into a 24-well plates (Nunc) containing gold disks coated with amino-terminated thiol (HS-C 11 -NH 3 Cl) purchased from Prochimia (Prochimia surfaces, Poland). Samples were incubated at 22  1°C for 1, 4 and 14 hours exposure times.

Fluorescence optical microscopy and membrane status assays

The bacteria shape and the integrity of the bacterial membrane were assessed using the BacLight TM stain kit (L7012, Molecular Probes, Eugene, USA). The cell-wall permeability in the absence and presence of the D-Ctl was examined according to a methodology described in previous works. [START_REF] Zaet | D-Cateslytin, a new antimicrobial peptide with therapeutic potential[END_REF][START_REF] Quiles | In situ and real time investigation of the evolution of a Pseudomonas fluorescens nascent biofilm in the presence of an antimicrobial peptide[END_REF] With this kit, bacteria with intact membranes exhibit green fluorescence (Syto 9), while bacteria with damaged membranes show red fluorescence (propidium iodide, PI). The bacterial suspensions were stained with the BacLight TM kit during 20 minutes in the dark. Then the suspension was filtered onto a 0.22 µm membrane, and the membrane was rinsed with non-pyrogenic sterile water to eliminate excess dyes. The sample was mounted in BacLight TM mounting oil as described by the instructions provided by the manufacturer.

Images were acquired with the × 100 oil immersion objective of an Olympus BX51 microscope equipped with an Olympus XC50 camera. Both fluorescences were observed using fluorescence filter cube U-MWIB3 (Olympus, excitation filter: BP 460-495 nm, emission filter: LP 510 nm).

ATR-FTIR spectroscopy

ATR-FTIR spectra were recorded on a Bruker Vertex70v spectrometer equipped with a KBr beam splitter and a DTGS detector. Spectra recording and data processing were performed using the Bruker OPUS 7.5 software. The resolution of the single beam spectra was 4 cm -1 .

Two hundred scans were collected per spectrum corresponding to a 2-minutes accumulation time. All interferograms were Fourier processed using the Mertz phase correction and a Blackman-Harris three-term apodization function. A nine-reflection diamond ATR accessory (DurasamplIR™, SensIR Technologies, incidence angle: 45°) was used for acquiring spectra.

No ATR correction was performed. ATR-FTIR spectra are shown with an absorbance scale corresponding to log(R reference /R sample ), where R is the internal reflectance of the device.

Measurements were performed at 22  1°C in an air-conditioned room. One drop of the bacterial suspension in LB medium was put on the ATR crystal. The spectral background was removed by recording the spectrum of the LB medium free of bacteria (obtained by centrifugation at 7000 g during 5 minutes). Water vapor subtraction was performed when necessary. All spectra were baseline corrected at 1800 and 900 cm -1 . FTIR measurements were performed at 22  1°C in an air-conditioned room.

Bacteria drying procedure for AFM imaging

Critical point drying (CPD) with CO 2 was used for sample preparation in order to get images at the nanoscale of the bacterial cells subjected or not to D-Ctl at various concentrations and for several exposure times. Indeed, this technique is one of the most accurate to preserve the highly fragile three-dimensional structure of biological samples especially when they are difficult to observe in aqueous media. [START_REF] Francis | Optimized sample preparation for high-resolution AFM characterization of fixed human cells[END_REF][START_REF] Kurtzman | Specimen holder to critical-point dry microorganisms for scanning electron microscopy[END_REF][START_REF] Dillow | Bacterial inactivation by using near-and supercritical carbon dioxide[END_REF] Gold slides incubated with bacterial samples were rinsed three times in baths of PBS for 5 minutes each. Then samples were dehydrated by immersion into 8 successive baths of water/ethanol mixtures (ratios of 90/10, 80/20, 70/30, 50/50, 30/70, 20/80, 10/90, 0/100) for 10 minutes each and stored in pure ethanol before CPD treatment. The CPD procedure is carried out using a homemade device [START_REF] Lazrag | Aerogel production by supercritical drying of organogels: Experimental study and modelling investigation of drying kinetics[END_REF] , and according to 5 successive steps: (1) samples are placed into an autoclave with 250 µL of pure ethanol onto each samples; (2) the autoclave is filled and pressurized with liquid CO 2 for 30 minutes up to 90 bars at 30°C; (3) the autoclave is heated for 20 minutes up to 45°C; (4) water and ethanol residues were extracted from the samples through the supercritical CO 2 (5) depressurization and decrease of temperature from 45°C to the ambient conditions during 30 minutes. At the end of these steps, the samples were taken out of the autoclave and the samples were stored in petri dishes tightly closed to prevent absorption of water in the samples.

Atomic Force Microscopy characterizations

Morphology images (dried samples) and in-situ monitoring of bacterial cells mechanical properties (living bacteria) were performed using the PeakForce tapping™ mode, with a Bioscope Resolve (Bruker Nano Surface, Bruker France SAS, Palaiseau, France). Silicon nitride cantilevers with conical tips were purchased from Bruker (NPG-10, Bruker France SAS, Palaiseau, France) with a spring constant of about 0.35 nN/nm. The applied force between the tip and the samples surface was carefully controlled and minimized at ~ 0.250 nN. All images were acquired with a resolution of 512 pixels by 512 pixels and a scan rate of 1.0 Hz. AFM images with scan sizes of 10 μm × 10 μm and 5 μm × 5 μm were then analyzed with Nanoscope Analysis for extracting roughness, thickness, and general morphology.

Mechanical properties of bacterial cells were investigated in LB medium using a MFP3D-BIO instrument (Asylum Research Technology, Oxford Instrument, Manheim, Germany) using silicon nitride cantilevers with conical tips (OMCL-TR400PSA-3, Olympus, Japan) with a spring constant of about 10-15 pN/nm. The bacterial stiffness was calculated by analyzing the force-indentation curves according to the Sneddon model. [START_REF] Sneddon | The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile[END_REF][START_REF] Gavara | Determination of the elastic moduli of thin samples and adherent cells using conical atomic force microscope tips[END_REF] In this model, the Young modulus is related to the applied force according to the equation given below:

    BECC 2 2 / 1 2 δ R ν 1 π α Tan 2 f E F     ( 1
)
where δ is the indentation depth,  the Poisson coefficient, α the semi-top angle of the tip and f BECC is the bottom effect cone correction function that take into account the presence of the substrate stiffness. [START_REF] Gavara | Determination of the elastic moduli of thin samples and adherent cells using conical atomic force microscope tips[END_REF] All mechanical measurements were determined using an automatic Matlab algorithm described elsewhere, [START_REF] Polyakov | Automated Force Volume Image Processing for Biological Samples[END_REF] and the average values given in this work were calculated from at least 2000 force curves.

Bacterial trapping and monitoring by fast scanning microscopy

Bacterial suspensions were injected through a syringe to a homemade nanofluidic chip containing bacterial traps [START_REF] Peric | Microfluidic bacterial traps for simultaneous fluorescence and atomic force microscopy[END_REF] and directly mounted on a custom integration of a Dimension Fast Scan AFM on an inverted optical microscope as described elsewhere. [START_REF] Odermatt | High-Resolution Correlative Microscopy: Bridging the Gap between Single Molecule Localization Microscopy and Atomic Force Microscopy[END_REF] By creating a pressure difference across the nanofluidic traps, the bacteria were attracted and physically immobilized into the traps for further AFM analyses. Morphology as well as mechanical properties of immobilized bacteria were obtained in LB medium using PeakForce QNM® mode using a ScanAsyst®-Fluid cantilever with 1.6 nN•nm -1 spring constant. During the bacterial monitoring procedure, the pressure in the microfluidic chamber underneath the membrane was kept between 20 and 100 mbar below atmospheric pressure.

Results and Discussion

Effect of D-Ctl on the bacterial growth of E. coli strains

The minimal inihibitory concentration (MIC) was determined at 8 µg/mL for the E2146 strain. [START_REF] Zaet | D-Cateslytin, a new antimicrobial peptide with therapeutic potential[END_REF] The MIC of E2152 was estimated in the same range, i.e. between 5 and 10 µg/mL in this work. Hereafter, MIC refers to the minimal inhibitory concentration for E2146. The growth of both bacteria at different initial OD 600 was monitored by measuring the OD 600 as a function of time for 24 hours (Figure 1). In absence of D-Ctl the middle of exponential growth phase is reached after 13-14, 9 and 9 hours for both strains at initial OD 600 set at 0.001, 0.1 and 0.5, respectively. The stationary growth phase begins after 24 hours for an initial OD 600 set at 0.001, and it was not reached for suspensions with initial OD 600 set at 0.1 and 0.5, only pseudo-plateaus are observed. The growth curves of bacteria incubated for 24 hours with D-Ctl at the MIC show several features as a function of the initial concentration of bacteria (Figure 1). At initial OD 600 set at 0.001, no growth of E2152 was recorded during 23 hours, whereas a significant bacterial growth started after 10 hours for strain E2146. At initial OD 600 set at 0.01, bacteria grew during ~ 6 hours and then a slightly descending plateau was reached, suggesting the stop of the multiplying process. When the initial number of bacteria was higher, i.e. DO 600 = 0.50, the OD 600 after 24 hours of culture were below those of control cultures. The bacterial growth continued at a slower rate than in control culture for strain E2152. In presence of D-Ctl at 5×MIC and 10×MIC, no significant bacterial growth was measured during 24 hours whatever the initial concentration of bacteria. These results demonstrated the existence of a concentration threshold from which D-Ctl totally inhibits the bacterial growth for a defined number of cells. Besides, the inhibitory and even the bactericidal activity of D-Ctl also depend on the initial number of bacterial cells exposed. To sum-up, these results strongly suggested that E2146 seems to tolerate D-Ctl better than 2152.

Bacterial membrane status and morphology under D-Ctl treatment

E2152 and E2146 bacteria were stained with the BacLight™ kit at the D-Ctl concentrations and exposure times previously described. Figure 2 and Figure 3 show representative images for E2152 and E2146, respectively. Except for few of them, the bacteria of both strains grown without D-Ctl were green showing rod-shaped bacteria with non-damaged membranes. The average lengths were estimated and are gathered in Table 1. Both bacteria strains have the same range of average length, i.e. ~2.3 to ~2.7 µm whatever the time of culture. Many bacteria were dividing, and here we did not discriminate bacteria just divided (length ~1.5 µm) from those that were dividing (lengths from ~2.0 to ~3 µm). 2b,f,j), and their shape and size did not change significantly with respect to bacteria not exposed to D-Ctl (control experiment). The average size was increased after 4 and 14 hours of exposure to D-Ctl at the MIC for red bacteria. The higher standard deviation for length measurements showed a non-homogenous behavior of the bacteria in front of D-Ctl (Table 1). The occurrence of a few filamentous cells can be noticed, and they were up to ~ 25 µm long after 14 hours of contact with D-Ctl. In addition to the increase of the permeability to PI, some bacteria were highly stressed showing it by the inhibition of the division of the cells. For concentrations of D-Ctl at 5×MIC and 10×MIC, all bacteria were red, showing damaged membranes. At 10×MIC and after 14 hours of contact, no bacteria were observed. This result suggested that almost all the bacteria were lysed by the action of D-Ctl or they were too much damaged to resist to the mechanical filtration. For 5×MIC and 10×MIC of D-Ctl, few bacteria become filamentous with lengths up to 18 µm, the average length of the remaining bacteria were close to those in absence of the peptide.

When E. coli E2146 was submitted to D-Ctl at the MIC, most of bacteria were red, but a nonnegligible number of bacteria remained green meaning that their membrane was not damaged (Figure 3b,f,j). The average shape and length of the green bacteria were close to each other and close to those observed for the bacteria in the control experiment whatever the time of exposure to D-Ctl. After 4 and 14 hours of contact with D-Ctl, red bacteria (i.e. with damaged membranes) had a higher average length with a high distribution of lengths, especially after 14 hours of contact with D-Ctl (4.1 ± 2.7 µm, Table 1, Figure 3f,j). A few bacteria developed long filamentous shapes, their length reached up to ~ 9 µm after 4 h and 24 µm after 14 hours of contact with D-Ctl indicating that the stress induced by the presence of D-Ctl was very high. Bacteria submitted to D-Ctl at 5×MIC and 10×MIC were almost all red, only a few of them were observed green for an exposure of 4 hours at 5×MIC. The average length increased slightly (less than 0.6 µm). In contrast of all the other conditions of D-Ctl exposure, the PI staining of E2146 at D-Ctl concentrations of 5 × MIC and 10 × MIC was not homogenous along the cell (Figure 3c,d,g,h,k,l). The fluorescence was highly concentrated at specific areas of the bacteria making to appear bright orange-red dots inside the bacteria.

Propidium iodide stains both RNA and DNA. This non-homogenous staining through the whole bacteria implies that nucleic acids were probably aggregated upon the presence of D-Ctl.

A closer look on the bacterial morphology was performed using AFM to decipher possible cell wall damages at the nanoscale. Several areas of 10 × 10 and 5 × 5 µm 2 were investigated and representative images were collected, and they are reported in Figure 4 and When E2152 cells are exposed to D-Ctl, the rod-shaped structure is maintained but the cell size was increased. Indeed, the average length and width are increased up to 2.3 ± 0.6 µm and 1.0 ± 0.2 µm, respectively. Such increase in size is also confirmed by the height profiles that evidenced bacterial diameters reaching from 650 nm up to 750 nm (Figure 6a). D-Ctl activity seems to inflate bacterial cells but no major damages as pores or important roughness have been observed with AFM even though the BacLight™ assays evidenced that almost all bacteria exposed to D-Ctl exhibited damaged membranes.

In absence of D-Ctl, E2146 exhibits a smooth and rod-shaped structure with an average length of about 1.4 ± 0.2 µm for a width of 0.7 ± 0.1 µm that are in accordance with epifluorescence images. The height profiles (Figure 6b) showed bacterial diameter of c.a. 600 nm. When E2146 cells are exposed to D-Ctl, an evolution of bacterial morphology similar to E2152 was observed. The rod-shaped structure is maintained but the cell size was increased with average length and width reached up to 2.5 ± 0.6 µm and 1.0 ± 0.3 µm, respectively. The analyses of height profiles (Figure 6) revealed an increase of the bacterial diameter from 600 nm up to 850 nm. As previously observed for E2152, bacterial cells seem to be also inflated without major detectable damages.

One can emphasize that permeabilization activity of D-Ctl is increased with concentration and exposure time but the rod-shaped structure of non-lysed treated cells is maintained whatever exposure time and D-Ctl concentration. In addition, the damages revealed by AFM could be associated to high membrane weakening that maybe enhanced by the drying process (Figure S10). Non-lysed treated cells are inflated upon D-Ctl exposure and the bacterial swelling is much higher in presence of Type 1 fimbriae. The membrane permeability and bacterial shape are intimately connected to the cell-wall mechanical features. In this way, the permeabilization or membrane activity of D-Ctl can be investigated through the elastic properties of the bacterial cell wall by force spectroscopy.

Evolution of bacterial stiffness during D-Ctl treatment

Bacterial stiffness of E2152 and E2146 was monitored during 24 hours under D-Ctl concentrations of 0 and 40 µg/mL (5 × MIC) and reported in Figure 7a,b. In absence of D-Ctl, Figure 7 shows that the bacterial stiffness of E2152 and E2146 remain very stable during 24 hours with values of c.a. 600 and 300 kPa, respectively. These values are in agreement with previous mechanical analyses of the two bacterial strains. [START_REF] Francius | Bacterial surface appendages strongly impact nanomechanical and electrokinetic properties of Escherichia coli cells subjected to osmotic stress[END_REF] When the bacterial cells are exposed to D-Ctl at 40 µg/mL, the bacterial stiffness shows two different behaviors according to the bacterial strains. Indeed, the bacterial stiffness of E2152 slightly increased from c.a.

600 kPa up to 900 kPa while it decreased from c.a. 300 kPa down to 40 kPa for E2146. Here, the impact of D-Ctl on the bacterial stiffness is highly influenced by the presence of the of Type 1 fimbriae structures. In the absence of Type 1 fimbriae, a stiffening of about 30% is observed over 24 hour-treatment whereas a dramatic softening of about 85% when Type 1 fimbriae are expressed.

The mechanical behavior of E2152 under D-Ctl treatment seemed counterintuitive in respect with the membrane permeabilization supported by the BacLight™ assays. However, mechanical stiffening under AMP has been already reported for huge peptides accumulation at bacterial envelope concomitantly to its permeabilization. [START_REF] Quiles | In situ and real time investigation of the evolution of a Pseudomonas fluorescens nascent biofilm in the presence of an antimicrobial peptide[END_REF][START_REF] Zhu | Rigidification of the Escherichia coli cytoplasm by the human antimicrobial peptide LL-37 revealed by superresolution fluorescence microscopy[END_REF][START_REF] Pogoda | Stiffening of bacteria cells as a first manifestation of bactericidal attack[END_REF] The mechanical behavior of E2146 under D-Ctl treatment is in good agreement with previous observations [START_REF] Zaet | D-Cateslytin, a new antimicrobial peptide with therapeutic potential[END_REF] that have evidenced a reduction of bacterial stiffness by a factor of 3 with respect to the untreated bacteria. This dramatic evolution of bacterial stiffness was explained by membrane permeabilization as supported by the BacLight™ assays and should reflect the loss of cytosol that led to bacterial lysis and death. [START_REF] Zaet | D-Cateslytin, a new antimicrobial peptide with therapeutic potential[END_REF][START_REF] Da Silva | Effects of the antimicrobial peptide PGLa on live Escherichia coli[END_REF][START_REF] Pott | Melittin modifies bending elasticity in an unexpected way[END_REF] Here, we clearly evidenced unexpected different To address these questions, and to understand the molecular origin in such differences of mechanical effects, the evolution of the biochemical fingerprints of D-Ctl and the two bacterial strains were monitored by Fourier transform infrared spectroscopy (ATR-FTIR).

Indeed, ATR-FTIR allows the investigation of physiological, biochemical and conformational changes of D-Ctl.

Infrared spectral fingerprints of the bacteria submitted or not to D-Ctl

IR-ATR spectra from both E. coli suspensions in presence or not of D-Ctl are shown in Figure 8 and Figure 9. One should notice that the background for every spectrum presented here is the supernatant free of bacteria corresponding to the suspension under study.

Therefore, the spectra are representative for the remaining bacteria only, the contribution of the medium being removed.

The spectra of the bacteria cultured without D-Ctl showed the specific features of Gramnegative bacteria. [START_REF] Zaet | D-Cateslytin, a new antimicrobial peptide with therapeutic potential[END_REF][START_REF] Freudenthal | Discrepancies between Cyclic and Linear Antimicrobial Peptide Actions on the Spectrochemical and Nanomechanical Fingerprints of a Young Biofilm[END_REF][START_REF] Quiles | Analysis of changes in attenuated total reflection FTIR fingerprints of Pseudomonas fluorescens from planktonic state to nascent biofilm state[END_REF] Usual signatures of the biochemical components of bacterial cells, i.e. proteins (P), nucleic acids (NA), phospholipids (PL) and polysaccharides (PS) are present (main assignments are given in Figures 8a and Figure 9a). The overall intensities of the whole spectra increased in as a function of the increase of the time of culture in accordance with the growth curves recorded with the same conditions (Figure 1).

When E2152 was submitted to D-Ctl for 1 and 4 hours, the general spectral signatures of the spectra were preserved with respect to those of the bacteria of the control experiment for all three concentrations of D-Ctl. However, the spectra stayed at the same level of intensities showing, in accordance with growth curves, that at least the growth was very quickly stopped (Figure 8). After 14 hours of D-Ctl exposure to bacteria, the signal-to-noise ratio of the spectra were very low, suggesting a very low concentration of bacteria in those suspensions.

After 14 hours of D-Ctl exposure, the intensities of the bands at ~1240 and 1085 cm -1 mainly assigned to PO 2 stretchings of nucleic acids were very low, suggesting that the metabolic activity was also very low.

One can emphasize that the contribution of PLs to the global spectral profile is low and the loss of the latter is negligible since no cell debris showing bacterial lysis has been observed either in AFM or in epifluorescence microscopy.

When E2146 was exposed to D-Ctl at the MIC, the general features of the spectra were similar at every time of exposure to D-Ctl, looking as those of bacteria from the control experiment (Figure 9b). In addition, the whole intensity of the spectrum after 1 hour of the D-Ctl exposure was similar to the control experiment after 1 hour of culture. The whole intensities of the spectra did not change drastically suggesting that bacteria did not grow upon the exposure to D-Ctl. However, bands assigned mainly to nucleic acids around 1230 and 1086 cm -1 , were present. They suggested that the metabolic activity was maintained at least partially for 14 hours at the MIC. The features of the spectra from the bacteria after contact with D-Ctl at 5 × and 10 × MIC were different (Figures 9c,d). Indeed, bands at ~1621 and ~1526 cm -1 newly appeared in the region of amide bands from proteins and peptides. These bands are assigned to -sheets. It has been shown that D-Ctl, which is unstructured in solution, forms antiparallel -sheets that aggregate at the surface of negatively charged bacterial mimetic membranes. [START_REF] Jean-Francois | Aggregation of cateslytin beta-sheets on negatively charged lipids promotes rigid membrane domains. A new mode of action for antimicrobial peptides?[END_REF] Therefore, it can be suggested that D-Ctl accumulated on or/and in the bacteria forming such aggregates on or/and in the bacteria. The spectral region between 1250 and 1000 cm -1 was very different with respect to the control spectra. The general shape of these bands, mainly assigned to PO 2 stretchings of nucleic acids, were less resolved as compared to the control spectra and their intensities relative to amide bands were lower (Figures 9c,d). The structure and the amount of nucleic acids were affected by the presence of D-Ctl. This suggested an intracellular action of D-Ctl when its concentration was very high. Altogether, spectroscopic analyses suggested that D-Ctl activity led to a dramatic decrease in metabolism of both bacterial strains until a total break depending on AMP concentration. Such interruption in bacterial metabolism seemed to be enhanced when bacterial cell-wall are decorated with type-1 fimbriae.

Impact of parietal structures on the D-Ctl mode of action

As regarding the results obtained from both microbiological and physico-chemical investigations, one can notice the occurrence of two different behaviors in D-Ctl activity as a function of the D-Ctl concentration (at MIC and ≥ 5 MIC). D-Ctl antibacterial activity is not related to the initial bacterial concentration considered for both strains, i.e. the concentrations are very similar whatever the strain for a given OD 600 . Besides, this activity is very different between the nude cell-wall and the Type 1 fimbriae decorated bacteria.

In details, when both bacterial strains are exposed to D-Ctl at MIC, almost all membranes are damaged after 4 and 14 hours without any pore formation upon AFM observations. However, the membranes are more impacted for the nude cell-wall bacteria than those with the Type 1 fimbriae. Besides, BacLight™ assays and ATR-FTIR measurements supported that their metabolism is not totally interrupted but disturbed. Indeed, E2146 bacteria appeared filamentous; the presence of D-Ctl disrupted the cell division. On the contrary very few cells of E2152 appeared filamentous. Both strains bacteria appeared inflated with higher diameter size and continued to elongate but failed to septate. This can be subsequent to the blocking of DNA replication, interruption in chromosome segregation, or inhibition of cell division. [START_REF] Salomon | Microcin 25, a novel antimicrobial peptide produced by Escherichia coli[END_REF][START_REF] Subbalakshmi | Mechanism of antimicrobial action of indolicidin[END_REF] One can notice that there is no significant evolution of bacterial stiffness at MIC concentration as evidenced in Figure 7d At 5×MIC and 10×MIC, both bacterial strains did not grow anymore. After 14 hours of exposure with D-Ctl, the ATR-FTIR spectroscopic investigations revealed that the amount of nucleic acids for E2152 is also low with respect to the protein amount, suggesting the stop of the metabolic activity. Concerning E2146, the cells have mostly damaged membranes.

However, the staining was not homogenous inside the bacteria, bright red-orange dots occurred within the cells. It is concomitant with aggregation of D-Ctl with the formation of sheets confirmed by ATR-FTIR investigations. As PI is a nucleic acids dye, one can suggest that the bacterial nucleic acids aggregated upon the binding of D-Ctl. Previous studies have shown that cationic AMP with antiparallel -sheets and/or rich in arginine can bind nucleic acids. [START_REF] Yonezawa | Binding of tachyplesin I to DNA revealed by footprinting analysis: significant contribution of secondary structure to DNA binding and implication for biological action[END_REF][START_REF] Park | Mechanism of action of the antimicrobial peptide buforin II: buforin II kills microorganisms by penetrating the cell membrane and inhibiting cellular functions[END_REF][START_REF] Sim | Investigating the nucleic acid interactions of histone-derived antimicrobial peptides[END_REF] The membrane is not the only target of the peptide as it is shown here for E2146.

Conversely, E2152 bacteria staining is homogeneous whatever the exposure time suggesting

Conclusion

In the present study, we report a systematic analysis of nanomechanical and spectrochemical properties related to the antibacterial mechanism of action on two E. coli strains expressing or not Type 1 fimbriae exposed to different D-Ctl doses and during several exposure periods. It is found that depending of D-Ctl concentration, bacterial stiffness is dramatically modified concomitantly to major membrane damages and important changes in the spectral fingerprint of the bacteria. However, no pore formation has been observed while all bacterial membranes were permeabilized under high D-Ctl concentrations. In case of the occurrence of fimbrial structures, an intracellular mode of action was shown also. Our results evidenced that D-Ctl activity is highly impacted by the cell-wall external structures and surface properties such as Type 1 fimbriae leading or not to macroscopic aggregation. In this way, as it was reported on the LPS modification, the modulation of parietal structures expressions could have a strong impact on the bacterial resistance/tolerance to antimicrobial peptides and therefore can be a strategy used by bacteria to bypass the immune defenses.

FIGURE AND TABLE CAPTIONS.

TABLES Table 1. Average length (µm) of bacteria in presence or not to antimicrobial peptide D-Ctl (red: average length for red bacteria in the images; green: average length for green bacteria in the images).

Figure 5 .

 5 In absence of D-Ctl, E2152 exhibits a smooth and rod-shaped structure with an average length of about 1.6 ± 0.4 µm for a width of 0.8 ± 0.1 µm, which are in accordance with epifluorescence images. The height profiles (FiguresS2 -S9) showed bacterial diameter of c.a. 650 nm.

  mechanical effects of D-Ctl on E. coli cells constitutively expressing or not the Type 1 fimbriae. How such opposite mechanical effects can be explained, and what is the influence of the bacterial external structures on the D-Ctl activity?

  Figure 7a, b.

  that D-Ctl remained at the bacterial cell-wall. The constitutive expression of Type 1 fimbriae with respect to the nude cell-wall strain can weaken the outer membrane and thus can facilitate the D-Ctl translocation. Therefore, D-Ctl activity could be characterized by the damaging of the outer membrane via electrostatic interactions followed by the peptide translocation. Further intracellular translocation results in the decrease or the inhibition of the acid nucleic synthesis. It is evidenced by the bacterial filamentation at the MIC mainly for E2146 and by the aggregation of nucleic acids at the highest concentrations of D-Ctl. The small differences observed in the bacterial growth for increasing concentration of D-Ctl, are supported by macroscopic effects such as modifications of the bacterial suspension turbidity. Indeed, Figure10showed drastic effects of D-Ctl increasing concentration on suspension turbidity and bacterial aggregation. For E2152 bacteria, the increase of D-Ctl concentration led to an important bacterial aggregation at the bottom of the spectrophotometer cuvettes delimited by the white dashed lines. For E2146 bacteria, the effect of increasing D-Ctl concentration is lower on bacterial aggregation and the bacterial suspension became less and less cloudy suggesting a more important bacteriolytic activity. These results are in agreement with the difference in bacterial surface charge density of the two strains. The surface of E2152 is almost 6 times more negatively charged than E2146 according to previous studies that highlighted surface charge density of about -170 mM and -30 mM for E2152 and E2146, respectively.[START_REF] Francius | Bacterial surface appendages strongly impact nanomechanical and electrokinetic properties of Escherichia coli cells subjected to osmotic stress[END_REF] Such differences should lead to larger interactions and accumulation of D-Ctl onto E2152 bacterial cell wall than the latter onto E2146. Further analyses combining zeta potential and dynamic light scattering could be more relevant to decipher the molecular mechanism behind this macroscopic phenomenon. Besides, overexpression of Type I fimbriae could significantly weaken the bacterial outer-membrane and facilitate its permeabilization by the antimicrobial peptides. This phenomenon has been previously reported for LPS modifications/expression modulation that impact on the bacterial resistance/tolerance to antimicrobial peptides and antibiotics.[START_REF] Band | Mechanisms of Antimicrobial Peptide Resistance in Gram-Negative Bacteria[END_REF][START_REF] Gunn | Bacterial modification of LPS and resistance to antimicrobial peptides[END_REF]

Figure 1 :

 1 Figure 1: Growth curves of E. coli MDR measured at 23°C by the recording of the optical

Figure 2 .

 2 Figure 2. Representative epifluorescence images of E. coli E2152 submitted or not to the presence of D-Ctl at different concentrations and several times of exposure (BacLight TM

Figure 3 .

 3 Figure 3. Representative epifluorescence images of E. coli E2146 submitted or not to the

Figure 4 .

 4 Figure 4. AFM imaging in air after CO 2 critical point drying of samples of E. coli E2152

Figure 5 .

 5 Figure 5. AFM imaging in air after CO 2 critical point drying of samples of E. coli E2146

Figure 6 .

 6 Figure 6. Morphology evolution of E. coli MDR exposed or not to D-Ctl at several

Figure 7 .

 7 Figure 7. In-situ monitoring of bacterial stiffness of E. coli MDR exposed or not to D-Ctl at 5

Figure 8 :

 8 Figure 8: ATR-FTIR spectra of planktonic E. coli E2152 submitted or not to the presence of

Figure 9 :

 9 Figure 9: ATR-FTIR spectra of planktonic E. coli E2146 submitted or not to the presence of

Figure 10 :

 10 Figure 10: Tentative mechanism of action of D-Ctl at the different concentrations on E2152

Table 1 :

 1 Average length (µm) of bacteria in presence or not to antimicrobial peptide D-Ctl (red: average length for red bacteria in the images; green: average length for green bacteria in the images).

		Time	Without D-Ctl	MIC	5×MIC	10×MIC
		1 hr	2.7 ± 0.6	2.9 ± 0.5	2.3 ± 0.3	2.6 ± 0.4
	E2152	4 hrs	2.7 ± 0.6	3.6 ± 1.0 (red) 2.6 ± 0.4 (green)	2.9 ± 0.8	2.5 ± 0.4
		14 hrs	2.6 ± 0.4	3.3 ± 0.8 (red) 2.6 ± 0.5 (green)	3.2 ± 0.5	-
		1 hr	2.3 ± 0.4	2.7 ± 0.2	2.9 ± 0.4	2.6 ± 0.4
	E2146	4 hrs	2.4 ± 0.4	2.9 ± 0.7 (red) 2.3 ± 0.3 (green)	2.6 ± 0.4	2.6 ± 0.3
		14 hrs	2.3 ± 0.4	4.1 ± 2.7 (red) 3.1 ± 1.7 (green)	2.6 ± 0.4	2.6 ± 0.4

When E. coli E2152 was exposed to D-Ctl at the MIC, almost all bacteria appeared red, and their average size slightly increased at all the contact times. Only a few of them were green showing intact membranes (Figures

Acknowledgements

The authors thank C. Beloin from Institut Pasteur -Paris for critical reading of the manuscript, fruitful help and discussion on microbiological aspects of our work. FQ and GF thank the Spectroscopy and Microscopy Service Facility (SMI) of LCPME (Université de Lorraine-CNRShttp://www.lcpme.cnrs-nancy.fr). GEF and OP acknowledge support by the Swiss national science foundation under grant number 205320_152675.

Supporting Information

Structures and physicochemical properties of D-cateslytin and scheme of bacterial strains cell wall (Figure S1), AFM imaging and height profiles of reference bacteria E2152 in absence of D-Ctl and performed in air after CO 2 critical point drying (Figure S2