Pedro Dias Longhitano
email: pedro.dias.longhitano@volvo.com

Christophe Bérenguer
email: christophe.berenguer@grenoble-inp.fr

Benjamin Echard
email: benjamin.echard.2@volvo.com

Battery degradation aware vehicle routing with speed and acceleration limitations

The problem of assigning missions to a fleet of electric vehicles has been thoroughly studied in the literature under the name of Electric Vehicle Routing Problem (EVRP), however most versions of such a problem do not take battery degradation into account. One of the main difficulties of battery degradation aware routing is the necessity of including State of Health (SoH) models that are designed for different operation conditions and the necessity of including a realistic State of Charge (SoC) model. In this paper we present a version of the EVRP that includes battery degradation as well as the inclusion of decision variables such as speed and acceleration limitations. We illustrate our approach through numerical experiments.

Introduction

Electric mobility has become increasingly important. With this new transport paradigm, it is crucial to account for battery degradation since this component can cost up to 50% of vehicle cost, [START_REF] Baghdadi | Lithium battery aging model based on dakin's degradation approach[END_REF], and has also important environmental consequences.

To achieve practical degradation aware routing, it is necessary to describe SoH evolution in the optimization problem, therefore, most of the known battery stress factors must be modeled and related to the performance of missions. As the most important stress factors of batteries are Depth-of-Discharge (DoD) and mean State of Charge (mSoC), [START_REF] Rosewater | Battery energy storage models for optimal control[END_REF], it is necessary to model the State of Charge (SoC) and connect it to vehicle usage. In most EVRP's, however, SoC models are oversimplified not accounting for the dynamics of the battery and variable terminal voltage, Pelletier et all (2019).

Furthermore, new technologies allow for different decisions to be considered. With today's connectivity, it is possible to change vehicle parameters in order to limit speed or accelerations. Those parameters directly influence usage, impacting battery degradation and enrich the optimization potential.

In this work, we propose and solve an optimization problem that accounts for battery degradation using realistic SoH and SoC models. In this optimization problem, not only route assignment is considered but also vehicle parameters changes, such as maximum speed and acceleration. Therefore, the main contributions of this paper can be summed up as:

 The formalization of an EVRP that accounts for battery degradation, maximum vehicle speed limitation, and maximum vehicle acceleration limitations.  A SoH estimation method suited for optimization problems that also includes realistic SoC modeling.  A set of numerical experiments to validate our approach. Accordingly, Section 2 presents the problem formalization, Section 3 shows the underlying SoH and SoC models, Section 4 presents numerical results and section 5 concludes the paper.

Problem Formulation

The goal of this paper is to derive a method to optimally assign missions to a fleet of electric trucks. Missions are seen as points in space, representing clients, that must be visited. Charging stations are also considered so that vehicles can recharge if needed. The mission plan must respect several constraints related to SoC range, vehicle capacity, and other logistical aspects that will be presented later on. All vehicles start at the same point, the headquarters, and must go back to it after finishing the missions.

To formalize this problem, we consider the graph G = (N,E) where nodes N represent every point in space that can be visited by vehicles. N = C ∪ S ∪ 0 where C is the set of clients (missions), S is the set of recharge stations and 0 is the headquarters. E represents the shortest paths between each node. The set of vehicles is denoted by K.

The decision variables of the problem are the following:

 x ijz which is equal 1 if vehicle z performs a displacement from node i to j.  w ijz which is the payload of vehicle z going from node i to j.  v max z which is the maximum allowed speed of vehicle z  a max z which is the maximum allowed acceleration of vehicle Z For the sake of notation, we define Xz = { x ijz = 1, ∀ i,j ∈ N} representing the mission plan of vehicle z. We also consider t ij (v max z ,a max z ) , the time necessary to go from node i to j with maximum allowed speeds and accelerations v max z and a max z , H, the maximum vehicle payload and SoC(t, X z , v max z , a max z ) which is the SoC at instant t of a vehicle following a mission plan X z with v max z and a max z . The optimization cost C is a combination of degradation, energy and delay costs and can be written as:

C = C deg + C energy + C delay (1)
The degradation cost is defined as:

C deg = ∑ z∈K c batt ΔSoH(X z ,v max z ,a max z )
(2) where c batt is a constant proportional to the battery cost and ΔSoH is the variation of SoH caused by following the mission plan X z with v max z and a max z . The energy cost can be written as:

C energy = ∑ z∈K ∑ i∈N ∑ j∈C c e e ijz (v max z ,a max z )x ijz (3) where c e is a constant representing the price of energy and e ijz (v max z ,a max z ) is the energy consumed going from node i to j with v max z and a max z . Finally, the delay cost is:

C delay = ∑ z∈K ∑ i∈N ∑ j∈C c d x ijz max[y ijz -l j ,0] (4
) The complete optimization problem can then be written as:

min C

(5) subjected to:

∑ z∈K ∑ j∈C x ijz = 1, ∀i ∈ N (6) ∑ j∈N x ijz = 1, ∀z ∈ K (7) ∑ j∈S x ijz ≤ 1 ∀i ∈ N,∀z ∈ K (8) ∑ z∈K ∑ j∈S x ijz -∑ z∈K ∑ j∈S x jiz = 0 ∀i ∈ N (9)
∑ z∈K ∑ j∈S w ijz -∑ z∈K ∑ j∈S w jiz = h i ∀i ∈ N (10)

h j x jiz ≤ w ijz ≤ (H -h j ) x jiz ∀i ∈ N,∀z ∈ K (11) y 0jz ≥ t 0j (v max z ,a max z )x 0jz ∀j ∈ C ∪ S,∀z ∈ K (12) ∑ z∈K ∑ j∈S y ijz -∑ z∈K ∑ j∈S y jiz = t ij (v max z ,a max z )x jiz ∀i ∈ N (13) min SoC(t,X z ,v max z ,a max z ) ≥ So C crit ,∀z ∈ K (14) x ijz ∈ {0,1} (15)
w ijz ≥ 0 (16)

y ijz ≥ 0 (17) v max z ∈ [ v - max , v + max ] (18)
a max z ∈ [ a - max , a + max ] (19
) Inequality (6) imposes that every client is visited once, Constraint (7) guarantees that the number of vehicles used does not exceed the size of the fleet, Constraint (8) limits the number of visits to recharge stations and Constraint (9) imposes that for each incoming arc there is a corresponding outgoing arc. Constraint (10) guarantees that, at each delivery, the payload is reduced according to h i which is the payload variation at client i ( h i = 0 if point i is the headquarter or a charging station), while Constraint (11) ensures that the maximum payload H is respected and it makes w ijz = 0 when x ijz = 0. Constraints ( 12) and ( 13) describe arrival time evolution, ensuring that the displacement duration is respected. Notice that in this formulation, charges are considered to happen instantly as well as deliveries. Constraint ( 14) imposes the range limitations through SoC.

Constraints ( 15) through ( 19) define the proper range of the decision variables. And a - max , a + max , v - max , v + max are the acceptable limits of a max z and v max z respectively. Additionally, in this model, we consider that anytime a recharge station is visited or a vehicle goes back to the headquarter, it will recharge its battery fully. The next section presents the models used to compute ΔSoH (X z ,v max z ,a max z ) , e ijz (v max z ,a max z ) and SoC(t, X z , v max z , a max z ) based on the decisions variables of the problem.

SoH and SoC estimation methods

SoH modeling

To compute the term ΔSoH (X z ,v max z ,a max z ) it is necessary to employ a model that can connect SoH variation to vehicle operation. This requires a degradation model that can be used for application with different usage conditions. We employ the model used by [START_REF] Xu | Modeling of lithium-ion battery degradation for cell life assessment[END_REF] in which SoH is expressed as:

SoH (t) = e -f d (SoC(t))
(20) with f d being defined as:

f d (SoC (t) ) = ∑ n(SoC(t)) i=1 ω i S σ S δ S T
(21) where n(SoC (t) ) is the number of SoC cycles that are obtained through the rainflow-counting algorithm Downing et al of the SoC signal up to time t. S δ , S σ and T are stress coefficients related DoD, mSoC and temperature and ω i is 1 in a full cycle or 0.5 in a half cycle. The stress factors coefficients are taken from [START_REF] Xu | Modeling of lithium-ion battery degradation for cell life assessment[END_REF] and can be expressed as:

σ i = e K σ (σ i -σ ref ) (22)
δ i = (K δ 1 δ δ 2 i + + K δ ) -1 (23)
T i = e K T (T i -T ref )T ref T i
(24) where K σ , K T , K δ 1 , K δ 2 , K δ are constants estimated through experiments, σ ref and T ref are experimental references. σ i and δ i are the cycle amplitude and cycle mean obtained through rainflow-counting and T i is the mean temperature of the cycle, considered to be normally distributed around the target temperature of the battery management system. Finally, the SoH variation is expressed as:

ΔSoH (X z ,v max z ,a max z ) = H 0 -SoH (X z ,v max z ,a max z )

where H 0 is the SoH before following a mission plan and (X z ,v max z ,a max z ) is the SoH obtained after performing mission plan X z with maximum speed and acceleration v max z , and a max z respectively 3.2 modeling Since the SoH model relies on the cycle decomposition of SoC(t), it is necessary to estimate while following a mission plan. Most EVRP's that are no concerned with degradation effects approximate SoC to be proportional to the travelled distance or consider SoC variation to be equal to energy consumption [START_REF] Pelletier | Battery degradation and behavior for electric vehicles: and numerical analyses of several models[END_REF].

To achieve realistic SoH estimation those approximations are limiting, and battery dynamics must be considered. Battery and vehicle dynamics can be combined to obtain SoC by solving the following system of algebraic differential equations: v = a(t, X z , v max z , a max z ) (26)

P mec (t) = ρ air C w A v(t) 2 +m g v (t)sin α +m g C r v (t)cos α +m a (t) v(t) (27)
P elec (t) = P mec (t) η(v(t),a(t))

(28)

I (t) = 1 2R ( V OC -V l (t) -V (t) -( V OC (SoC(t)) -V l (t) -V (t)) 2 -4 R 0 P elec (t) ) (29) V = - V s R C + I(t) C (30) V l = - V l R l C l + I(t) C l (31) = - I(t) Q 0 (32)
Equation ( 26) is the cinematic relation of vehicle speed v(t) and acceleration a(t, X z , v max z , a max z ). Equation ( 27) expresses the mechanical power as a function of vehicle mass m, gravity acceleration g, road inclination α, C r is the rolling resistance coefficient and C w is the drag coefficient and A is the frontal area of the vehicle. Equation ( 28) expresses the electrical power with η(v (t) ,a (t) ) being the power train efficiency which is a function of instantaneous speed and acceleration. Equation ( 29) to (32) represent the dynamic of the battery that is modeled as a second order circuit. R l , R 0 and R s are the resistances of the equivalent circuit, C l and C s the capacitances and Q 0 the nominal capacity of the battery.

For the system to be solved, it is necessary to estimate the function a(t, X z , v max z , a max z ) and the function Voc(Soc). In this work, the acceleration function is estimated using the same method of Longhitano et al ( 2022) and Basso et al (2019). It consists of using road information of the path defined by X z , such as the presence of traffic lights, crossroads and maximum allowed speed, road stretches lengths, and the maximum speed, v max z and acceleration, a max z . Voc(Soc) can be obtained empirically or is made available by battery suppliers as a look-up table.

With those equations, the decisions variables of the problem can be used to define the acceleration function and determine the SoC profile which in turns define the variation of SoH. Energy can be computed simply by integrating the electrical power.

Results

Degradation effect on Routing

To validate the interest of considering degradation on the optimization model we perform numerical on the grid shown in Figure 1. In those experiments, first we generate different instances of the problem (a set of random clients, C random charging stations, S). Each instance is solved three times: once considering c batt = 0, which is equivalent to classical EVRP formulations in which only energy consumption is minimized, once considering c e = 0 which is equivalent to minimizing only battery degradation and finally once with c batt ≠ 0 and c e ≠ 0.

Table 1 shows the different routes found solving the optimization problem in the different cases and 1 illustrate the different routes found for instance 1.

Due to the computational complexity added by the non-linearity of the degradation model and the differential equations of SoC, those instances are using genetic techniques. can be seen, including battery degradation changes the optimal mission plan. When accounting for degradation, more recharges are added to the plan. They are not necessary from the point of view of ensuring a minimum SoC level, however they reduce DoD and consequently, degradation. Whenever a vehicle recharges, it stops and needs to accelerate again to leave it, thus spending more energy. Our optimization model can handle this trade-off and find better solutions than the one found by considering only energy consumption or only battery degradation. Figure 2 shows the SoC trajectories of the different solutions for instance 1. It shows that the extra charges indeed reduce SoC amplitude and even if they introduce extra cycles, since their amplitude is smaller, they create less damage. Accounting for degradation leads to cost improvement in all instances. It is important to highlight that even in instances where the cost reduction is subtle, since SoH evolves slowly, small reductions in degradation add up through time and can lead to several extra months of operations.

Instance

Speed and acceleration effect

To isolate the effect of the parameters v max z and a max z a second set of experiments is performed. Instances of the problem are generated randomly. Each instance is solved twice, once with v max z and a max z as decision variables and once with them set as v + max and a + max . Results are shown in Table 2.

Table 2: Solutions of different instances of the problem considering the possibility of affecting maximum vehicle speed and acceleration.

As can be seen, limiting speeds and accelerations has a positive impact on the overall cost. This is a direct consequence of equation 27. The cost reduction is a reflex of less energy consumption but also less degradation, since SoC cycles will have smaller amplitudes because of affecting such parameters.

Conclusion

This paper presented a variation of the EVRP that accounts for degradation and limitation on speeds and accelerations. To make it useful for degradation minimization in realistic scenarios, the optimization model is built upon a realistic SoC estimation method. Since both the degradation and SoC models are computationally challenging to incorporate in optimization problems, a genetic algorithm was used to solve it. Through numerical experiments, the importance of realistic SoC modeling, degradation-aware decision-making and limiting speeds and accelerations is shown. Incorporating degradation and extending the set of possible actions of the problem yields to better solutions the most optimization models. real life applications, this can have a beneficial impact since it can extend the useful life of vehicles and reduce the long-term exploitation cost. future works, it is necessary to compare the proposed model with real vehicle data, both for SoC and SOH calibration. Incorporating stochastic elements is also desirable. example, traffic randomness can highly affect energy consumption and delays and therefore, be accounted for. Long-term routing methods are also interesting paths for future research. From an algorithmic point of view, developing new optimization techniques for this problem is also an interesting research path since genetic algorithms do not necessarily converge to a global minimum and due to the computational complexity, performance is still a limiting factor for solving bigger instances.

Number

Figure 1 .

 1 Routes found for instance 1. The red route indicates the solution when accounting for degradation and energy consumption (Case A). The green route indicates the solution of the classic EVRP version where only energy is considered. The blue route is the solution considering only degradation.

Figure 2 .

 2 Figure 2. SoC trajectories for different routes found for instance 1.

Table 1 :

 1 Solutions different instances of the problem considering only energy consumption, only battery degradation and both. The symbol * indicates recharging stations.

		c batt	c e	Mission plan	Cost C
		€	€		€
	1	10000 0.18	A0, E0, E1*, E3, D5, A4, A2, A0	8.60
	1	0	0.18	A0, E0, E3, D5, A4, A2, A0	8.63
	1	10000	0	A0, E0, E1*, E3, D4*, D5, A4, A2, A0	8.62
	2	10000 0.18	A0, E0, E1*, E3, E5, A4, A2, A0	9.55
	2	0	0.18	A0, E0, E3, E5, A4, A2, A0	9.60
	2	10000	0	A0, E0, E1, E3, E5, D4*, A4, A2, A0	9.57
	3	10000 0.18	A0, C5, F5, F2, D2*, A2, A0	9.56
	3	0	0.18	A0, C5, F5, A2, A0	10.52
	3	10000	0	A0, C5, F2, D2*, A2, A1*, A0	9.68
	4	10000 0.18	A0, B3, D3*, F3, C0, A0	7.65
	4	0	0.18	A0, C0, A0	7.73
	4	10000	0	A0, B3, D3*, F3, C1*, C0, A0	7.68

 Longhitano, P.D., Tidriri, K., Bérenguer, C., Echard, B., 2022.Joint optimization of routes and driving parameters for battery degradation management in electric vehicles, in: 11th IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes-SAFEPROCESS 2022, p. 557-562, 2022. Rafael Basso, Balazs Kulcsar, Bo Egardt, Peter Lindroth, andIvan Sanchez-Diaz. Energy consumption estimation integrated into the electric vehicle routing problem. Transportation Research Part D: Transport and Environment, 69:141-167, 2019.


domain: spi phys

text:
J P Colombier

E Moreno

A Rudenko

V Fedorov

Polarization and complexity effects on ultrafast laser-induced nanoscale surface structuring

Ultrafast laser pulses confine a large number of photons spatiotemporally, allowing them to sculpt solid surfaces with structures smaller than 100nm. Upon multi-shot irradiation the self-organization and growth of periodic patterns arise from localized perturbations of the optical coupling on random surface nanoreliefs. This study shows how simulating engineered ultrafast laser beams interacting with rough surfaces provides unrivaled optical control for creating suitable artificial media with nanofeatures that outperform those of naturally occurring surfaces, with promised novel applications in biomedicine, nanocatalysis, or metaphotonics.

INTRODUCTION

The modification and structuring of materials using ultrafast femtosecond laser pulses is a mature technology that has opened up wide fields of research and numerous applications. Today, these focus on the extreme manufacture of surfaces with nanocavities, peaks, grooves or periodic surface stripes. Technological breakthroughs consist in obtaining new structures formed in synergy with the hydromechanical response of the material, which can have a high aspect ratio, or by shapes arranged asymmetrically by non-conventional polarization. Also, temporal and polarization pulse shaping allows the response of the excited material to be controlled along specific relaxation processes and can result in exotic shapes or symmetries.

METHODS

The key concept of the approach lies in the control of polarization, vorticity, duration, and transverse energy distribution along the light propagation up to the interaction with an inhomogeneous surface. Currently, numerical simulations of laser-surface interaction relies on Finite-Difference Time-Domain schemes that require to initialize the beam from a predetermined intensity distribution and given polarization states [START_REF] Moreno | Axicon lenses with chiral-focusing properties modeling by means of analytical functions[END_REF][START_REF] Moreno | Full Explicit Numerical Modeling in Time-Domain for Nonlinear Electromagnetics Simulations in Ultrafast Laser Nanostructuring[END_REF]. In this work, vectorially polarized fields or asymmetrical and twisted beams are used as boundary conditions of the Maxwell equation to generate complexstructured light fields with specially designed intensity, phase, and polarization distributions [START_REF] Zhang | Laser-induced periodic surface structures: Arbitrary angles of incidence and polarization states[END_REF]. Along with the experimental achievements, we have developed a model combining three-dimensional electromagnetic and hydrodynamic approaches to deliver a comprehensive frame for understanding light-induced structuring on tens of nanometer scales [START_REF] Rudenko | High-frequency periodic patterns driven by non-radiative fields coupled with Marangoni convection instabilities on laser-excited surfaces[END_REF]. The numerical calculations allow us to investigate the multipulse evolution of surface relief in a self-consistent way, evaluating both the inhomogeneous absorption due to surface roughness and the thermomechanical flows.

RESULTS

From a fundamental perspective, complex surface topography is driven by the regulation that originates from the constructive and destructive roles of the electromagnetic and hydrodynamic processes. During the lifetime of a transient laser-induced melting layer, we uncover that dissipative structures can emerge and continuously exchange energy and entropy with photon flux to ensure relief growth and topographical organization during liquid flow. Self-organized nanopatterns can be frozen by solidification, demonstrating the potential of ultrafast laser process for the fabrication of advanced surfaces with a topography designed at the nanoscale. Upon multi-pulse excitation, the electromagnetic roles of collective effects and inter-pulse feedback on the resulting surface topographies will be presented owing to 3D numerical approach [START_REF] Rudenko | Self-organization of surfaces on the nanoscale by topography-mediated selection of quasi-cylindrical and plasmonic waves[END_REF]. This calculation efficiently couples electromagnetism to hydrodynamics in order to simulate the laser energy distribution, the hydrodynamic flows and the propagation of shock/rarefaction wave.

The presented results on multiphysics simulation deliver a comprehensive frame for the understanding of light-induced structuring on record scales as we elucidate that the generation of nanopatterns result from the interplay between non-radiative optical response on surface, liquid flows driven by surface wave excitation and thermomechanical relaxation. Any surface exposed to sufficiently intense laser irradiation tends to organize along the stationary electromagnetic fields due to the excitation of scattered and surface plasmon waves at the surface. The interference patterns constitute a light footprint that has been observed and exploited since the 70's. The scale of this patterning is mainly defined by the laser wavelength and the features can be controlled by the electric field polarization and incident angle. In addition to this light coupling, the surface self-organizes in response to random fluctuation of light absorption. This establishes complexity fingerprints sculpted on the surface. Driven by optical near fields triggering convection process in a transient thin liquid film, the structures appear at the nanoscale (< 100 nm periodic features) and carry the stigma of nonlinear dynamics.

CONCLUSION

Upon ultrafast multiple pulse laser irradiation, morphology-mediated feedback effects strengthen the energy deposition where the electromagnetic response is enhanced. This can be simulated by photonics computations for advanced vectorial beams, revealing complex energy distribution on a surface. The optical and then hydrodynamic response of the rough surface now offers new ways to manipulate material properties at the nanoscale as topographical structures spontaneously emerge with high potential applications in nanotechnologies.

Figure 1 :

 1 Figure 1: (a) Schematic illustration of numerical electric field distribution for ultrafast laser propagation. (b) Interaction with a rough surface. (c) Inhomogeneous absorption of the field. (d) Nanostructure growth by feedback at different number of pulses (experimental).


Charlène Cornilleau

Laura Alvarez

Christine Wegler

Cyrille Billaudeau

Felipe Cava
email: felipe.cava@umu.se

Rut Carballido-López
email: rut.carballido-lopez@inrae.fr

Peptidoglycan remodeling in response to cell wall acting antibiotics inBacillus subtilis

Keywords: Peptidoglycan, cell-wall, antibiotic, Bacillus subtilis

come

Introduction

The bacterial cell wall (CW) is a complex, semi-rigid extracellular meshwork that maintains shape and provides structural integrity to bacterial cells. The principal constituent of the CW is peptidoglycan (PG, also known as the murein sacculus): a three-dimensional network of linear sugar strands of alternating N-acetyl-glucosamine (GlcNAc) and N-acetyl-muramic acid (MurNAc) cross-linked by stem peptides attached to MurNAc. The stem peptides are synthesized as pentapeptide chains and contain the unusual D-amino acids. In most Gram-positive species the pentapeptide chain is typically made of Lalanine (L-Ala) in position 1, D-glutamine or D-iso-glutamic acid (D-Glu) in position 2, L-lysine in position 3 and a C-terminal D-alanine-D-alanine (D-Ala-D-Ala) dipeptide [START_REF] Strange | The Isolation, Characterization and Chemical Synthesis of Muramic Acid[END_REF][START_REF] Mandelstam | The Incorporation of Amino Acids into the Cell-Wall Mucopeptide of Staphylococci and the Effect of Antibiotics on the Process[END_REF][START_REF] Vollmer | Peptidoglycan Structure and Architecture[END_REF][START_REF] Pazos | Peptidoglycan ». In Bacterial Cell Walls and Membranes[END_REF]. However, in Bacilli as well as in Mycobacteria and Gram-negative species, the lysine in position 3 is substituted by meso-2,6diaminopimelic acid (m-DAP), leading to the following stem peptide sequence for the Gram-positive model bacterium Bacillus subtilis: L-Ala, D-Glu, m-DAP, D-Ala, D-Ala.

PG synthesis results from a series of sequential reactions that occur in three overall stages. First, the soluble, activated nucleotide PG precursors UDP-GlcNAc and UDP-MurNAc-pentapeptide are synthesized in the cytoplasm by the sequential action of the Mur enzymes. Then, the soluble precursors are transferred to undecaprenyl phosphate (UP, also referred as C55 lipid carrier) at the inner leaflet of the membrane to finally form the lipid-anchored GlcNAc-MurNAc-pentapeptide monomer subunit, namely lipid II, which is then flipped across the membrane. Last, externalised lipid II is polymerised into a nascent glycan chain by transglycosylation (TG) and, concomitantly, crosslinked to the existing sacculus by transpeptidation (TP). Figure 1 displays the PG synthetic pathway and known enzymes in B. subtilis. Historically, most of the enzymes responsible for TG and TP reactions were discovered because of their ability to bind penicillin, and were thus called penicillin-binding proteins (PBPs). PBPs are classified depending on their activities: Class A and B high-molecular weight PBPs are PG synthases, and Class C low molecular-weight PBPs are PG hydrolases. Class A PBPs (aPBPs) possess two functional domains and perform both TG and TP reactions, while Class B PBPs (bPBPs) are monofunctional and only have a TP catalytic domain. In addition to aPBPs, the penicillin-insensitive essential RodA and FtsW proteins also catalyze TG from lipid II, working with their cognate bPBPs TPs to synthesize sidewall and septal PG, respectively [START_REF] Emami | RodA as the Missing Glycosyltransferase in Bacillus Subtilis and Antibiotic Discovery for the Peptidoglycan Polymerase Pathway[END_REF][START_REF] Taguchi | FtsW Is a Peptidoglycan Polymerase That Is Functional Only in Complex with Its Cognate Penicillin-Binding Protein[END_REF]. TP reactions by aPBPs and bPBPs cleave the terminal D-Ala from a pentapeptide and form a peptide bridge between the D-Ala in position 4 of the acyl-donor peptide and the D-center of the m-DAP in position 3 of the acyl-acceptor stem peptide (Fig. 1). This 43 crosslink, termed D,D-transpeptidation (DD-TP), strictly requires a pentapeptide as substrate [START_REF] Terrak | The Catalytic, Glycosyl Transferase and Acyl Transferase Modules of the Cell Wall Peptidoglycan-Polymerizing Penicillin-Binding Protein 1b of Escherichia Coli[END_REF]) and is the main crosslinking strategy in most bacteria including B. subtilis [START_REF] Warth | Structure of the peptidoglycan from vegetative cell walls of Bacillus subtilis[END_REF][START_REF] Atrih | Analysis of Peptidoglycan Structure from Vegetative Cells of Bacillus Subtilis 168 and Role of PBP 5 in Peptidoglycan Maturation[END_REF][START_REF] Vollmer | Peptidoglycan Structure and Architecture[END_REF].

In addition to PG synthases, growth of the sacculus requires the activity of PG hydrolases, which cut existing bonds to allow the insertion of new material and thus the expansion of the PG meshwork without increasing its thickness [START_REF] Typas | From the Regulation of Peptidoglycan Synthesis to Bacterial Growth and Morphology[END_REF]. PG hydrolases can cleave almost any glycoside and peptide bond and belong to three different families depending on their substrate: (i) glucosaminidases, muraminidases, lytic transglycosylases, and also the lysozymes produced by many organisms as a major effector of their innate immune system, cleave the sugar backbone, (ii) amidases cut the bond between MurNAc and the stem peptide, and (iii) peptidases hydrolyze peptide bonds, either within or between peptide stems (Vollmer et al. 2008;[START_REF] Hashimoto | Digestion of Peptidoglycan near the Cross-Link Is Necessary for the Growth of Bacillus Subtilis[END_REF]. Peptidases include Class C low molecular-weight PBPs (cPBPs) and can be further classified as carboxypeptidases (CPs) or endopeptidases (EPs) depending on whether they remove the carboxy-terminal amino acid of the stem peptide or cleave non-terminal peptide bonds, respectively. Peptidases can also be classified depending on their stereospecificity, whether they cleave the bond between two D-chiral centers (DD-EPs and DD-CPs) or between and L-and a D-chiral center (LD-and DL-EPs and CPs) (Fig. 1). PG crosslinking and the potential lethal activity of PG hydrolases (autolysis) are also regulated by covalent modifications of their substrate (either the glycan strands or the stem peptides). Common PG modifications known to modulate PG hydrolases include O-acetylation of MurNAc residues [START_REF] Guariglia-Oropeza | Bacillus Subtilis σ V Confers Lysozyme Resistance by Activation of Two Cell Wall Modification Pathways, Peptidoglycan O-Acetylation and D -Alanylation of Teichoic Acids[END_REF][START_REF] Yadav | Bacterial Strategies to Preserve Cell Wall Integrity against Environmental Threats[END_REF] and also of GlcNAc in some species [START_REF] Moynihan | O-Acetylated Peptidoglycan: Controlling the Activity of Bacterial Autolysins and Lytic Enzymes of Innate Immune Systems[END_REF][START_REF] Bernard | Dual Role for the O-Acetyltransferase OatA in Peptidoglycan Modification and Control of Cell Septation in Lactobacillus Plantarum[END_REF]), de-N-acetylation of GlcNAc [START_REF] Atrih | Analysis of Peptidoglycan Structure from Vegetative Cells of Bacillus Subtilis 168 and Role of PBP 5 in Peptidoglycan Maturation[END_REF][START_REF] Yadav | Bacterial Strategies to Preserve Cell Wall Integrity against Environmental Threats[END_REF] and de-amidation of D-Glu or m-DAP [START_REF] Vollmer | Peptidoglycan Structure and Architecture[END_REF][START_REF] Dajkovic | Hydrolysis of Peptidoglycan Is Modulated by Amidation of Meso -Diaminopimelic Acid and Mg 2+ in Bacillus Subtilis: Homeostasis of Peptidoglycan Hydrolysis in Bacillus Subtilis[END_REF]. The detailed composition of the vegetative sacculus of B. subtilis was reported just before the end of the last century and emphasized the de-amidation of m-DAP and the de-Nacetylation of GlcNAc as main PG modifications, and showed a role in PG maturation for the major DD-CP PBP5 (encoded by dacA) [START_REF] Atrih | Analysis of Peptidoglycan Structure from Vegetative Cells of Bacillus Subtilis 168 and Role of PBP 5 in Peptidoglycan Maturation[END_REF].

PG modifications are part of a complex regulatory response and underlay the plasticity of the PG sacculus to adapt to a changing environment and/or to face conditions challenging its structural integrity. Among the latter, antibiotics targeting the CW are on the front line. Because of its essentiality and also because it is specific to bacteria, the bacterial CW and in particular PG synthesis represent the most prominent target of antibiotics. Almost every step of PG synthesis is targeted by at least one known antibiotic [START_REF] Schneider | An Oldie but a Goodie -Cell Wall Biosynthesis as Antibiotic Target Pathway[END_REF] (Fig. 1). For example, fosfomycin blocks the first committed cytoplasmic step of PG synthesis by covalently binding to the active site of MurA, which inactivates the enzyme [START_REF] Aghamali | Fosfomycin: Mechanisms and the Increasing Prevalence of Resistance[END_REF][START_REF] Dörries | Impact of Antibiotics with Various Target Sites on the Metabolome of Staphylococcus Aureus[END_REF][START_REF] Kahan | The Mechanism of Action of Fosfomycin (Phosphonomycin)[END_REF]. D-cycloserine is a structural analogue of D-Ala and targets different enzymes that use D-Ala as substrate [START_REF] Neuhaus | The Enzymatic Synthesis of D-Alanyl-d-Alanine. Iii. on the Inhibition of d-Alanyl-d-Alanine Synthetase by the Antibiotic d-Cycloserine *[END_REF][START_REF] Lambert | Mechanism of D-Cycloserine Action: Alanine Racemase from Escherichia Coli W[END_REF]. It inhibits Alr and Ddl, synthesizing the D-Ala-D-Ala dipeptide [START_REF] Walsh | Enzymes in the D-Alanine Branch of Bacterial Cell Wall Peptidoglycan Assembly[END_REF][START_REF] Batson | Inhibition of D-Ala:D-Ala Ligase through a Phosphorylated Form of the Antibiotic D-Cycloserine[END_REF]. Recently, it was proposed that D-cycloserine can also inhibit transpeptidases [START_REF] Kuru | Mechanisms of Incorporation for D -Amino Acid Probes That Target Peptidoglycan Biosynthesis[END_REF]. Tunicamycin blocks PG synthesis by competing with UDP-M5 to bind MraY [START_REF] Liu | The Membrane Steps of Bacterial Cell Wall Synthesis as Antibiotic Targets[END_REF][START_REF] Noda | TmrB Protein, Responsible for Tunicamycin Resistance of Bacillus Subtilis, Is a Novel ATP-Binding Membrane Protein[END_REF], and it has also been shown to inhibit TG/TP reactions [START_REF] Ramachandran | Screen for Inhibitors of the Coupled Transglycosylase-Transpeptidase of Peptidoglycan Biosynthesis in Escherichia Coli[END_REF]. Bacitracin is an antimicrobial peptide that binds to the UPP released upon PG precursor TG, preventing its dephosphorylation and thus inhibiting the recycling of the lipid carrier [START_REF] Radeck | Anatomy of the Bacitracin Resistance Network in B Acillus Subtilis: Anatomy of the Bacitracin Resistance Network[END_REF][START_REF] Stone | Mechanism of Action of Bacitracin: Complexation with Metal Ion and C55-Isoprenyl Pyrophosphate[END_REF]. Vancomycin is also an antimicrobial peptide that binds the terminal D-Ala-D-Ala dipeptide of the externalised PG precursors, preventing TG and TP by steric hindrance [START_REF] Mainardi | Evolution of peptidoglycan biosynthesis under the selective pressure of antibiotics in Gram-positive bacteria[END_REF]. Finally, the beta-lactam ring of penicillin G (and beta-lactam compounds in general) mimics the D-ala-D-ala dipeptide, inhibiting PG crosslinking by blocking DD-TPs and DD-CPs [START_REF] Schneider | An Oldie but a Goodie -Cell Wall Biosynthesis as Antibiotic Target Pathway[END_REF].

Although the primary targets of CW antibiotics are known, the events resulting from primary target inhibition and leading to either adaptation or cell death remain largely unknown. Among these, changes in the chemical composition and structure of the sacculus induced by CW antibiotics might reflect mechanism modulating CW properties. Given the rising emergence of multi-resistant bacteria, it is imperative to comprehensively study the effect of antibiotics in model organisms and use this knowledge for the development of new tools to combat pathogenic bacteria. Here, we have performed the first comprehensive analysis of the effect of PG-targeting antibiotics in the composition and structure of the bacterial CW. Using a high-throughput approach for the preparation and analysis of PG samples [START_REF] Hernandez | Genome-Wide Peptidoglycan Profiling of Vibrio Cholerae[END_REF], we studied the changes in the sacculus of B. subtilis in response to exposure to lysozyme and to six antibiotics targeting different steps of the PG biosynthetic pathway at different sub-lethal concentrations and growing conditions. Notably, our study reveals that cells adapt to the presence of PG synthesis inhibitors: after long exposure times, cultures reach similar optical densities as untreated cultures and restore their pool of intracellular soluble PG precursors.

However, the amount of PG in the walls of antibiotic-treated cells is strongly reduced and its structure significantly altered: sacculi are less cross-linked, more N-acetylated and less amidated, suggesting that the adaptative response involves activation of PG hydrolases. We also found interesting correlations between crosslinking and both de-N-acetylation and amidation of dimers. Remarkably, we additionally observed that the dose-dependent decrease in crosslink produced by TP inhibitors was correlated with the accumulation of de-amidated disaccharide-tripeptide monomer subunit (M3). Analysis of individual mutants affected in different aspects of PG metabolism provided some insight into the de-N-acetylation mechanism and uncovered a novel pathway for muropeptide de-amidation. Importantly, our study highlights a key role for endopeptidases in CW antibiotics tolerance and provides additional insights into the mechanisms of PG homeostasis.

Results and discussion

Experimental design

To gain insight into the mechanisms regulating PG homeostasis, we aimed at analysing the muropeptide profile of B. subtilis in response to a comprehensive set of antimicrobials targeting different steps of PG synthesis (Fig. 1). The synthesis of soluble PG precursors was inhibited by fosfomycin and D-cycloserine. Synthesis of the lipid-linked PG precursors was inhibited by tunicamycin and bacitracin. Note that tunicamycin primarily blocks WTAs synthesis but at the high concentrations used here (> 10 µg/ml) it additionally blocks PG synthesis [START_REF] Brandish | Slow Binding Inhibition of Phospho-N-Acetylmuramyl-Pentapeptide-Translocase (Escherichia Coli) by Mureidomycin A[END_REF][START_REF] Campbell | Synthetic lethal compound combinations reveal a fundamental connection between wall teichoic acid and peptidoglycan biosyntheses in Staphylococcus aureus[END_REF].

Polymerisation and crosslinking of the PG meshwork was targeted by vancomycin and penicillin G. We also analysed the effect of the hydrolytic enzyme lysozyme, and used the antibiotic chloramphenicol, which blocks protein synthesis by targeting ribosomes but does not impact cell surface expansion [START_REF] Kitahara | Cell-Envelope Synthesis Is Required for Surface-to-Mass Coupling, Which Determines Dry-Mass Density in Bacillus Subtilis[END_REF], as negative control. We aimed to investigate the direct responses as well as the adaptation of B. subtilis to CW targeting antibiotics. To this end, cells were treated with sub-lethal concentrations of the compounds either at early-exponential growth (OD600 ≈ 0.2) during 2h ('2h protocol') or at very low density (OD600 ≈ 0.01) during 17h ('overnight protocol') (Fig. S1A) (See M&M for the detailed procedures). Seven increasing sublethal concentrations of each compound were used for each protocol for muropeptide analysis (Fig. S1, Fig. S2 and Fig. S3). We also extracted and quantified the soluble PG precursors from cultures treated with one low and one high antibiotic concentration (Fig. S1).

Cells adapt to grow in the presence CW antibiotics with little amount of PG

Addition of PG synthesis inhibitors to actively growing cells had a quick impact in growth rate. Upon addition of the compounds, cultures stopped growing for 15-30 min and then often lysed for 30-120 min before resuming growth again for all compounds excepting tunicamycin (Fig. 2A, Fig. S2C-J top panels). The concentrations of tunicamycin used to inhibit PG synthesis in this study strongly inhibit WTAs synthesis as well, which may explain the different behaviour. For all other compounds, an adaption to grow with an impaired PG biosynthetic pathway was ultimately implemented and cells started growing again. Viability and/or growth defects of several mutants affected in different aspects of CW synthesis can be rescued by millimolar concentrations of magnesium in the growth medium [START_REF] Formstone | A Magnesium-Dependent MreB Null Mutant: Implications for the Role MreB in Bacillus Subtilis: MreB of B. Subtilis[END_REF][START_REF] Chastanet | The Actin-like MreB Proteins in Bacillus Subtilis: A New Turn[END_REF], possibly through an inhibition of autolysins by Mg ++ [START_REF] Dajkovic | Hydrolysis of Peptidoglycan Is Modulated by Amidation of Meso -Diaminopimelic Acid and Mg 2+ in Bacillus Subtilis: Homeostasis of Peptidoglycan Hydrolysis in Bacillus Subtilis[END_REF][START_REF] Tesson | Magnesium Rescues the Morphology of Bacillus Subtilis MreB Mutants through Its Inhibitory Effect on Peptidoglycan Hydrolases[END_REF]). When we supplemented the cultures with 20 mM MgSO4, lysis was inhibited and/or growth rate improved (Fig. S2; bottom panels), suggestive of unbalanced PG hydrolytic activity in the presence of the PG inhibitors. For penicillin G-, vancomycinand tunicamycin-treated cells, the compensatory effect of Mg ++ was very mild and only effective at the lowest concentrations of the antibiotics (Fig. S2F-H), indicative of a stronger or different deregulation of PG hydrolases in these conditions. After 2h treatment with CW antibiotics, the amount of PG per cell ranged between 17% and 80% of the untreated control, and only the chloramphenicol treated control cells contained normal amounts of PG (Fig. 2B, Fig. S4). For antibiotics inhibiting the synthesis of the intracellular soluble precursors, the concentration of UDP-M3 and UDP-M5 species was consistent with the antibiotic target (Fig. 2C and Table S1A). In fosfomycin-treated cells, UDP-M3 was strongly depleted. Bacitracin, by inhibiting the recycling of lipid carrier, induced the accumulation of UDP-M5 (1.7 to 6-fold). Meanwhile, D-cycloserine -by inhibiting the formation of the D-Ala-D-Ala dipeptide-induced an impressive accumulation of UDP-M3, up to 320-fold for the high concentration.

Inhibition of lipid I synthesis by tunicamycin brought about the expected accumulation of UDP-M5 and to a lesser extent of UDP-M3. As previously reported [START_REF] Vemula | Antibiotic Effects on Methicillin-Resistant Staphylococcus Aureus Cytoplasmic Peptidoglycan Intermediate Levels and Evidence for Potential Metabolite Level Regulatory Loops[END_REF], antibiotics targeting TG and TP reactions also affected the concentration of intracellular soluble muropeptides. Vancomycin produced a massive accumulation (118-fold) of UDP-M5 [START_REF] Cegelski | Rotational-Echo Double Resonance Characterization of the of Vancomycin on Cell Wall Synthesis in Staphylococcus Aureus[END_REF][START_REF] Singh | Hidden mode of action of glycopeptide antibiotics: inhibition of wall teichoic acid biosynthesis[END_REF]and also an important accumulation of UDP-M3 (13-fold) (Fig. 2C and Table S1A). By blocking newly externalised lipid II, vancomycin may prevent recycling of the lipid carrier to which UDP-M5 should be transferred to form new lipid I and II, leading to accumulation of UDP-M5 in the cytoplasm, and thus of the upstream UDP-M3 too. Penicillin G also induced an accumulation of both UDP-M5 and UDP-M3 (Fig. 2C and Table S1A). Interestingly, antibiotic-induced accumulation of the intracellular soluble UDP-M5 and/or UDP-M3 precursors affected mostly their non-amidated forms (Table S1). Since all lipid II is externalised in its amidated form [START_REF] Atrih | Analysis of Peptidoglycan Structure from Vegetative Cells of Bacillus Subtilis 168 and Role of PBP 5 in Peptidoglycan Maturation[END_REF], this is consistent with the intracellular membranebound PG precursors being the preferential substrate for m-DAP amidation [START_REF] Levefaudes | Diaminopimelic Acid amidation in Corynebacteriales[END_REF][START_REF] Vollmer | Peptidoglycan Structure and Architecture[END_REF].

Addition of the antibiotics to low density cultures induced a longer (dose-dependent) lag phase after which cells grew in the presence of the antibiotic and reached a similar final optical density to untreated cultures (Fig. 2D, Fig. S3). Tunicamycin was again the only exception: it drastically reduced both growth rate and final optical density reached (Fig. 2D, Fig. S3F). The total amount of PG was still more strongly reduced for all compounds with the exception of bacitracin-treated cells, which had a normal PG amount (Fig. 2E, Fig. S4). In B. subtilis, bacitracin is known to induce the expression of several resistance modules that jointly maintain CW homeostasis [START_REF] Radeck | Anatomy of the Bacitracin Resistance Network in B Acillus Subtilis: Anatomy of the Bacitracin Resistance Network[END_REF]. After 17 h of growth in the presence of the antibiotics, the amount of intracellular soluble precursors was however back to normal for most antibiotics (Fig. 2F and Table S1B), consistent with an adaptation of cells to grow with an impaired PG synthetic pathway. Only UDP-M3 and UDP-M5 remained significantly accumulated in D-cycloserine-and tunicamycin-treated cells, respectively, but to a lesser extent than after 2h treatment (compare Fig. 2C andF).

Taken together, these results indicate that the effects of short exposure to PG synthesis inhibitors mainly reflect primary target inhibition. First, cultures stop growing upon inhibition of the PG synthetic step. PG precursors upstream the blocked step accumulate and the total amount of PG is reduced.

Second, cells start lysing, presumably because the balance between PG synthesis and PG hydrolysis is disrupted as suggested by the recovery observed in the supplementation with Mg ++ . However, cells ultimately adapt to grow in the presence of the antibiotics with a strongly reduced amount of PG in their walls.

Sacculi of cells growing in the presence of PG inhibitors are less crosslinked but have more trimers

To get a better understanding of adaptation to grow in the presence of CW antibiotics, we performed a detailed analysis of muropeptide composition. Ultraperformance liquid chromatography (UPLC) analysis coupled to mass spectrometry (MS) allowed the identification of the main muropeptide species in our B. subtilis PG profile (Fig S5A). The main resolved monomeric species included disaccharide-dipeptide (M2) and -tripeptide (M3) and M3 amidated in the m-DAP (M3-NH2).

Disaccharide-pentapeptide with one amidation in the m-DAP (M5-NH2) co-eluted with the M2 muropeptide. Based on MS quantification, M5-NH2 contributes to only 0.2% of the peak (M2: 99.8%) in untreated cells. All main dimers contained 43 crosslinks (resulting from DD-TP) and were of type D43 (disaccharide tetrapeptide -donor-with the terminal D-Ala linked to the carboxyl terminal m-DAP of a disaccharide tripeptide -acceptor-), with one or two m-DAP amidation(s) and de-N-acetylated or not, as previously reported [START_REF] Warth | Structure of the peptidoglycan from vegetative cell walls of Bacillus subtilis[END_REF][START_REF] Atrih | Analysis of Peptidoglycan Structure from Vegetative Cells of Bacillus Subtilis 168 and Role of PBP 5 in Peptidoglycan Maturation[END_REF]). The muropeptides eluting in peaks 7 and 8 have the same mass and amino acid composition (D43-NH2) (Fig. S5A). We believe these are isoforms and the difference in retention is due to the position of the amide group, on either m-DAP of the dimer. Additionally, the mass of peak 9 ([M+1] =1790.8) could correspond to a D43 with 4 amidations (D43-4*NH2; mass error of 18.9 ppm). MS fragmentation did not allow us to identify where the two putative additional amidations are, possibly on the D-Glu (yielding D-isoglutamine) as this is common in many species [START_REF] Warth | Structure of the peptidoglycan from vegetative cell walls of Bacillus subtilis[END_REF][START_REF] Vollmer | Peptidoglycan Structure and Architecture[END_REF]. However, we did not detect any monomer (M3-NH2) amidated on the D-Glu, and if there was any monomer doubly amidated (M3-2*NH2) it would most likely have eluted too early in our chromatogram profiles to be detected. Finally, two major trimeric species of type T443 (disaccharide tetrapeptide disaccharide tetrapeptide disaccharide tripeptide), with two or three amidations (T443-2*NH2 and T443-3*NH2, respectively) were also resolved (Fig. S5A).

Based on the quantification of these muropeptide species for each antibiotic condition (Fig. S5B-C and Table S2), we first assessed overall structural features and covalent modifications of the PG (Fig. 3A).

For all PG inhibitors, monomers importantly accumulated while dimers importantly decreased, in particular after overnight treatment, reducing the cross-linking of the sacculus and suggesting reduced TP and/or increased activity of EPs. Sacculi were greatly enriched in monomers resulting from both DD-EP and DL-EP activity (Fig. 3A and Fig. 4A-B). Another noticeable feature was accumulation of trimers after 2h treatment with PG precursors inhibitors and after overnight treatment with bacitracin and TP/TG inhibitors. Trimers are thought to have a relevant structural role as they are hubs connecting multiple glycan strands [START_REF] Vollmer | Peptidoglycan Structure and Architecture[END_REF]. They have been suggested to represent the linking point of short-to-long glycan strands [START_REF] Vollmer | Peptidoglycan Structure and Architecture[END_REF], and also to help strengthening the sacculus when glycan strands are short ( [START_REF] Pedro | Structural Constraints and Dynamics of Bacterial Cell Wall Architecture[END_REF]. Since cells grown in the presence of TP/TG inhibitors have down to 10% of PG relative to untreated cells (Fig. 2E and Fig. S4B), cross-linking of trimers may help to ensure sacculus integrity. Trimers are more compact than dimers and thus they may also be less accessible to the activity of EPs X) (de [START_REF] Pedro | Structural Constraints and Dynamics of Bacterial Cell Wall Architecture[END_REF].

Adaption to CW antibiotics involves a reduction in de-N-acetylation and amidation of the sacculus

We also assessed changes in amidation and N-acetylation, which have been shown to participate in CW homeostasis by modulating the autolytic activity of PG hydrolases [START_REF] Moynihan | O-Acetylated Peptidoglycan: Controlling the Activity of Bacterial Autolysins and Lytic Enzymes of Innate Immune Systems[END_REF][START_REF] Yadav | Bacterial Strategies to Preserve Cell Wall Integrity against Environmental Threats[END_REF][START_REF] Dajkovic | Hydrolysis of Peptidoglycan Is Modulated by Amidation of Meso -Diaminopimelic Acid and Mg 2+ in Bacillus Subtilis: Homeostasis of Peptidoglycan Hydrolysis in Bacillus Subtilis[END_REF]. Note that O-acetylation was not detected in our analysis. Our PG extraction protocol involves treating samples at basic pH (8.5-9) for reduction and then adjusting to acidic pH (3-4) [START_REF] Alvarez | Ultra-Sensitive, High-Resolution Liquid Chromatography Methods for the High-Throughput Quantitative Analysis of Bacterial Cell Wall Chemistry and Structure[END_REF]. Since O-acetyl groups are labile when subjected to basic or acidic pH, they were likely stripped out (Guariglia-Oropeza et Helmann 2011). Amidation of m-DAP and N-acetylation of GlcNAc occur intracellularly, presumably at the level of m-DAP of lipid II [START_REF] Atrih | Analysis of Peptidoglycan Structure from Vegetative Cells of Bacillus Subtilis 168 and Role of PBP 5 in Peptidoglycan Maturation[END_REF][START_REF] Vollmer | Peptidoglycan Structure and Architecture[END_REF][START_REF] Dajkovic | Hydrolysis of Peptidoglycan Is Modulated by Amidation of Meso -Diaminopimelic Acid and Mg 2+ in Bacillus Subtilis: Homeostasis of Peptidoglycan Hydrolysis in Bacillus Subtilis[END_REF]. Newly externalised lipid II precursors are 100% amidated and N-deacetylated [START_REF] Vollmer | Peptidoglycan Structure and Architecture[END_REF][START_REF] Hoyland | Structure of the ldcb ldcarboxypeptidase reveals the molecular basis of peptidoglycan recognition[END_REF]) and enzymes with de-amidase and de-acetylase activity regulate these modifications in the sacculus. De-N-acetylation (de-Ac) abolishes binding of LysM domains to polysaccharides containing GlcNAc including PG [START_REF] Mesnage | Molecular Basis for Bacterial Peptidoglycan Recognition by LysM Domains[END_REF]. Since LysM domains are often found in PG hydrolases, it is thought that de-Ac reduces their binding to PG, and thus their activity [START_REF] Rae | Mutations of the Listeria Monocytogenes Peptidoglycan N-Deacetylase and O-Acetylase Result in Enhanced Lysozyme Sensitivity, Bacteriolysis, and Hyperinduction of Innate Immune Pathways[END_REF][START_REF] Yadav | Bacterial Strategies to Preserve Cell Wall Integrity against Environmental Threats[END_REF]. m-DAP amidation was shown to allow physiological control over PG hydrolysis in B. subtilis; it was proposed that decreasing the amount of doubly amidated dimers in the sacculus may allow to increase the activity of one or more PG hydrolases [START_REF] Dajkovic | Hydrolysis of Peptidoglycan Is Modulated by Amidation of Meso -Diaminopimelic Acid and Mg 2+ in Bacillus Subtilis: Homeostasis of Peptidoglycan Hydrolysis in Bacillus Subtilis[END_REF][START_REF] Tesson | Magnesium Rescues the Morphology of Bacillus Subtilis MreB Mutants through Its Inhibitory Effect on Peptidoglycan Hydrolases[END_REF].

The sacculi of untreated cells were less de-N-acetylated and less amidated during exponential growth (2h protocol) than in stationary phase (overnight protocol) (Fig. 3BD and Fig. S6A, open symbols, and Table S2), consistent with the key role of PG hydrolases in active CW growth. In the presence of CW antibiotics, changes in these two covalent modifications were more important after overnight treatment (Fig. 3A), suggesting that they may be part of the adaptative response to the presence of the antibiotics. Sacculi were less de-N-acetylated for all antibiotic treatment, while total amidation (of all muropeptide species: monomers, dimers and trimers) was strongly reduced in penicillin G and vancomycin-treated cells. Interestingly, the amount of singly amidated dimers was only slightly affected by most antibiotics, while the amount of doubly amidated dimers was significantly reduced, especially in cells grown overnight in the presence of tunicamycin, vancomycin and penicillin G (Fig.

3A).

When we pooled together the results for all concentrations of each antimicrobial compound, for the 2 protocols, two correlations emerged. First, crosslink was correlated with total de-Ac, mostly after overnight adaptation (Fig. S6A-B). This was expected since de-Ac is calculated as the relative amount of de-N-acetylated muropeptides, and this modification was only detected in dimers in our chromatograms (Fig. S5A). Since total crosslink was calculated as dimers+2*trimers, the two parameters are directly proportional. When we calculated instead the percentage of de-Ac within dimers (% of de-Ac dimers relative to all dimers), a correlation was still found after overnight adaptation (Fig. S6C-D), in particular in the presence of vancomycin and penicillin G (TP inhibitors) (Fig. 3B and Fig. S6E-I). Such correlation might suggest that cells growing in the presence of TP inhibitors reduce de-Ac to -counter-intuitively-upregulate PG hydrolases, which further reduce crosslinking.

Alternatively, reduced de-Ac could be a consequence of reduced crosslinking. In this scenario, de-Ac could be functionally related to TP, with TP by PBPs preceding de-Ac of dimers. Then, if TP is inhibited de-Ac would be reduced. To test this hypothesis, we analysed the muropeptide composition of TP mutants including ΔponA (encoding PBP1, the major aPBP of B. subtilis), previously reported to have a sacculus less crosslinked than wild-type cells [START_REF] Atrih | Analysis of Peptidoglycan Structure from Vegetative Cells of Bacillus Subtilis 168 and Role of PBP 5 in Peptidoglycan Maturation[END_REF], ΔpbpH and ΔpbpA (encoding the coessential PbpH and PBP2A, respectively, major bPBPs). The sacculi of ΔponA mutant cells were less crosslinked and, as hypothesized, contained less de-N-acetylated dimers relative to the wild-type (Fig. 3C). The PG of the ΔpbpH mutant was more crosslinked, possibly because of deregulation of PBP2A, and dimers significantly more de-N-acetylated. However, dimers were also more de-N-acetylated in the ΔpbpA mutant but its crosslinking was similar to the wild-type (Fig. 3C). Whether the TP and de-Nacetylation reactions are functionally coupled will require further investigation.

We also found a significant correlation between crosslinking and amidation of the sacculus, in particular amidation of dimeric muropeptides, for all antibiotics and mostly upon overnight adaptation too (Fig. 3D-E). The reduction in crosslinking was correlated with a reduction in both singly and doubly amidated dimers for all antibiotics (Fig. S7). By de-amidating dimers, cells are thought to increase the activity of PG hydrolases [START_REF] Dajkovic | Hydrolysis of Peptidoglycan Is Modulated by Amidation of Meso -Diaminopimelic Acid and Mg 2+ in Bacillus Subtilis: Homeostasis of Peptidoglycan Hydrolysis in Bacillus Subtilis[END_REF][START_REF] Tesson | Magnesium Rescues the Morphology of Bacillus Subtilis MreB Mutants through Its Inhibitory Effect on Peptidoglycan Hydrolases[END_REF]. Based on these results, it is tempting to speculate that reduced crosslinking could be a consequence increased PG hydrolytic activity, and required for growth in the presence of PG synthesis inhibitors.

Antibiotics inhibiting TP reactions induce a strong accumulation of non-amidated M3

To gain insight into muropeptides de-amidation and a possible upregulation of PG hydrolases in the presence of CW antibiotics, we analysed in detail the variations of individual muropeptide species. The heatmap in Figure S8 further shows that long-term exposure to PG synthesis inhibitors triggers a modification in muropeptide composition more pronounced than a 2h treatment. The most striking variation was the accumulation of non-amidated M3 in most conditions (Fig. S8 and Table S2). This increase was particularly important in cells treated overnight with lysozyme, vancomycin or with high doses of penicillin G, which also accumulated M2 significantly (Fig. 4A-B, Fig. S8 and Table S2). Matrices of correlation between the different datasets showed that PG inhibitors in general produced a correlated accumulation of peak 3 (coelution of M2 and M5-NH2) and M3 monomers that anticorrelated with changes in the amount of M3-NH2 (Fig. 4C and Fig. S9). Furthermore, we found an inverse correlation between crosslink and M3 for all PG synthesis inhibitors and conditions (Fig. 4D), while no significant correlation was found between crosslink and M3-NH2 (Fig. 4E), indicating that M3 specifically accumulates when peptide crosslinks are reduced. This correlation was primarily given by the compounds that more strongly affected crosslink: lysozyme, penicillin G and vancomycin (Fig. S10A), which also brought about the strongest reduction in amount of PG in the sacculus (Fig. 2E).

Penicillin G and vancomycin inhibit TP/DD-CP and TG/TP reactions, respectively, on the newly externalised PG precursor, uncrosslinked amidated disaccharide pentapeptide (M5-NH2) (Fig. 4F), which would be expected to accumulate in these conditions. M5-NH2 co-eluted with M2 in our chromatograms (99.8:0.2 M2:M5NH2 in untreated cells) (Fig. S5, peak 3) and the corresponding peak increased in penicillin G-and vancomycin-treated cells (Fig. 4B).

Since penicillin G blocks both DD-TPs and DD-CPs (Fig. 4F), the observed accumulation of M3 might result from either (a) de-amidation of M5-NH2 into M5 by a yet unknown de-amidase followed by hydrolysis of M5 directly into M3 by a LD-EP (CwlK), or by residual DD-CP (PBP5, PBP5*, PBP4a, DacF) [START_REF] Smith | Autolysins of Bacillus Subtilis: Multiple Enzymes with Multiple Functions[END_REF] and then LD-CP (LcdB), or (b) hydrolysis of M5-NH2 into M3-NH2 first followed by de-amidation of M3-NH2 into M3 by the same set of enzymes. To further investigate this, we analysed the muropeptide profile of a ΔdacA (Fig. S11). dacA encodes PBP5, the major DD-CPase of B. subtilis [START_REF] Atrih | Analysis of Peptidoglycan Structure from Vegetative Cells of Bacillus Subtilis 168 and Role of PBP 5 in Peptidoglycan Maturation[END_REF] and in its absence cells greatly accumulate M5-NH2, which becomes the dominant muropeptide (Fig. S11) [START_REF] Atrih | Analysis of Peptidoglycan Structure from Vegetative Cells of Bacillus Subtilis 168 and Role of PBP 5 in Peptidoglycan Maturation[END_REF]). Interestingly, de-amidated M5 was detected in ΔdacA mutant cells (Fig. S11 and Table S2), indicating that de-amidation can occur in uncrosslinked pentapeptide stems (Fig. 4G). However, M3 accumulated less than 2-fold in the ΔdacA mutant (Fig. S11), while in penicillin-treated and vancomycin-treated cells it accumulated up to 2.8 and 4.4-fold, respectively, in the same conditions. Thus, accumulation of M5 did not correlate with an accumulation of M3, further suggesting that M5 is not the main transient intermediate of M3. The strong anticorrelation found between M3-NH2 and M3 monomers (Fig. 4C) supports the idea that accumulation of M3 results from de-amidation of M3-NH2 into M3. We concluded that inhibition of TP reactions either diverts the consumption of M5-NH2 into M3-NH2 by the action of CPs or LD-EPs and M3-NH2 is then de-amidated into M3 or, alternatively, that accumulation of M3 in these conditions results from increased DD-EP activity on dimers. In the latter case, the M3-NH2  M3 step may still be at play since only M3 accumulated while M3-NH2 decreased, yet both are expected products of DD-EPs.

It would be interesting to analyse PG composition of dacA, ldcB and cwlP mutants (Fig. 4F,G) in the presence of TG/TP inhibitors to conclude about the origin of M3.

Conclusions

Our study revealed that cells adapt to the presence of CW antibiotics in an unexpected way: they reduce their PG amount, decrease the degree of de-N-acetylation and of amidation of dimers. This suggests that an increase of PG hydrolase activity is required for cell survival in presence of PG inhibitors. Finally, the decrease in PG crosslinking triggered by CW antibiotics correlates with an accumulation of M3, giving us some insight about the mechanisms at play during cell adaptation to PG inhibitors. These findings highlight a cellular response to primary target inhibition that leads to adaptation and tolerance to grow in the presence of PG inhibitors.

Materials and Methods

General procedures and growth conditions

Bacillus subtilis 168 wild-type strain was grown in LB medium at 30°C for overnight cultures and at 37°C for day cultures. Growth protocols are detailed in the results section. The ∆ponA, ∆dacA ,∆pbpA, ∆pbpH, ∆ldcB mutants are from the Bacillus Knock-Out Kanamycin library [START_REF] Koo | Construction and Analysis of Two Genome-Scale Deletion Libraries for Bacillus Subtilis[END_REF].

Antibiotics

Stock solution of antibiotics were prepared as follows: fosfomycin 10 mg/mL in H2O, bacitracin 10 mg/mL in H2O, D-cycloserine 10 mg/mL in H2O, tunicamycin 10mg/mL in H2O with a few drops of 1 M NaOH, penicillin G 10 mg/mL in H2O, vancomycin 1 mg/mL in H2O, lysozyme 10 mg/mL in H2O, chloramphenicol 10 mg/mL in ethanol.

Growth protocols & growth curves

Two growth protocols were used in the study (Fig. S1A). For the "2h protocol" an overnight culture grown in LB at 30°C was diluted in LB to ODD600 0.01 and incubated at 37°C in a plate reader with reading of optical density at 600 nm until mid-exponential phase (OD ~ 0.2, t0); drug was added at t0 and cells were incubated again in the plate reader for 2h before collection. For the "ON protocol", a pre-culture was realized during the day in LB at 37°C and diluted to ODD600 0.01 in LB; drug was added at the same time as dilution (t0) and growth was carried O.N. (~17h) at 37°C in a plate reader with reading of optical density at 600nm. Two independent replicates (cultures, extraction and quantification) were realized for each condition.

Analysis of the PG of the mutant strains was done in triplicate; the wild-type and mutants were grown in LB until exponential phase when the samples were taken.

Intracellular soluble precursor preparation

Sample preparation of intracellular soluble PG samples was performed as follows [START_REF] Hernández | Modulation of Peptidoglycan Synthesis by Recycled Cell Wall Tetrapeptides[END_REF]. Cells were grown according to the "2h protocol" and the "overnight protocol' described above; OD was measured prior to sample collection for further normalization of the results. 500µL of cells were then centrifuged and pellets washed twice in 500 µL 0.9% NaCl before freezing. Pellets were resuspended in 100 µL water and autoclaved for 20 min at 120°C. After centrifugation the supernatant was recovered and filtered. Soluble muropeptides were detected and characterized by MS/MS analysis as follows. Identity of the peaks was confirmed using a Xevo G2-XS QTof system (Waters Corporation, USA). The instrument was operated in positive ionization mode. Detection of muropeptides was performed by MS E to allow for the acquisition of precursor and product ion data simultaneously, using the following parameters: capillary voltage at 3.0 kV, source temperature to 120 ºC, desolvation temperature to 350 ºC, sample cone voltage to 40 V, cone gas flow 100 l/h, desolvation gas flow 500 l/h and collision energy (CE): low CE: 6 eV and high CE ramp: 15-40 eV. Mass spectra were acquired at a speed of 0.25 s/scan. The scan was in a range of 100-2000 m/z. Data acquisition and processing was performed using UNIFI software package (Waters Corp.). An in-house compound library built in UNIFI was used for detection and identification of soluble muropeptides. Quantification was performed by integrating peak areas from extracted ion chromatograms (EICs) of the corresponding m/z value of each muropeptide.

Soluble precursors analysis was in biological duplicates.

Muropeptide preparation

High-throughput PG isolation, muropeptide separation and quantification were realized essentially as described in [START_REF] Hernandez | Genome-Wide Peptidoglycan Profiling of Vibrio Cholerae[END_REF]. Briefly, sacculi were isolated by resuspension of 500µl cell pellet in LB with 1.6% SDS and autoclaving 15 min at 120°C and 1 atm. Boiled sacculi were transferred to 96well-filter plates (AcroPrep Advance 96-well 0.2 μm GHP membrane filter plates, 2 ml volume, PALL) and washed 3 times with 800 µL water; covalently bound proteins were removed by adding 200 µL Tris 100mM pH8 and 0.4µL of proteinase K to the filters and incubating 30 min at 37°C. After an additional wash with 700µL water and centrifugation, the filter plate was transferred to a clean deepwell collecting plate. PG digestion was realised in situ with 1µL of muramidase at 1mg/mL in 50µL of water, O.N. at 37°C. Eluted muropeptides were collected in the collection plate and submitted to sample reduction. pH was adjusted to 8.5-9 by addition of 8µL borate buffer 0.5M pH9 and treated with 25µL NaBH4 2M for 30min RT. Reaction was stopped and pH adjusted to 2.0-4.0 with orthophosphoric acid 25%.

Muropeptides were analyzed by UPLC, as previously described [START_REF] Alvarez | Analysis of Gram-negative bacteria peptidoglycan by ultra-performance liquid chromatography[END_REF]. Briefly, muropeptides were separated using an UPLC system (Waters) equipped with a trapping cartridge precolumn (SecurityGuard ULTRA Cartridge UHPLC C18 2.1 mm, Phenomenex) and an analytical column (Waters ACQUITY UPLC BEH C18, 130Å, 1.7 µm, 2.1 mm×150 mm), using as solvents 0.1% (v/v) formic acid in Milli-Q water (buffer A) and 0.1% (v/v) formic acid in acetonitrile (buffer B) in a 14 min run. Muropeptides were detected by measuring the absorbance at 204 nm. Peak identities were confirmed by MS/MS as described above.

Muropeptide analysis was performed in biological duplicates for each condition.

Chromatographic data analysis and quantifications

Collected UPLC chromatographic data were analyzed using the chemometric-based pipeline described in [START_REF] Kumar | PG-Metrics: A Chemometric-Based Approach for Classifying Bacterial Peptidoglycan Data Sets and Uncovering Their Subjacent Chemical Variability ». Édité par Ivo G. Boneca[END_REF]. Raw data were trimmed and baseline corrected. Then, data sets were aligned by segments using the COW algorithm.

The total PG amount was calculated as the sum of all peak areas, and normalized using the OD600 at the time of collection. Relative values were calculated using the untreated sample as reference.

The relative area of each muropeptide was calculated by dividing its peak-area by the total area of the chromatogram.

The different PG features were calculated as follows: Total EP products = peaks 1+2+3 Amidation = % of amidated species vs all possibly amidated species = 100* (peak1 + 0.002*peak3 + 2*peak4 + peak5 + 2*peak6 + peak7 + peak8 + 2*peak9 + 2*peak10 + 3*peak11) / ((peak1 + peak2 + 0.002*peak3) + 2*(peak4 + peak5 + peak6 + peak7 + peak8 + peak9) + 3*(peak10 + peak11))

Amidation Dimers = 100* (2*peak4 + peak5 + 2*peak6 + peak7 + peak8 + 2*peak9) / (2*(peak4 + peak5 + peak6 + peak7 + peak8 + peak9))

Doubly amidated dimers / all dimers =100* (peak4 + peak6 + peak9) / (peak4 + peak5 + peak6 + peak7 + peak8 + peak9)

Singly amidated dimers/ all dimers = 100*(peak5 + peak7 + peak8) / (peak4 + peak5 + peak6 + peak7 + peak8 + peak9)

Data analysis and statistical methods

At least two independent experiments of each condition were considered and used to calculate mean and standard deviation (error bars), except for the scatterplot showing correlations (Fig. 3B, Fig. S6, S7, Fig. 4D,E) where all replicates are plotted. On these scatterplots, R 2 of correlation was calculated using linear regression. Heatmaps (Fig. 3A, Fig. S8) realized using GraphPad software.

Correlation matrices

Correlations were calculated using R (version 4.1.1) in the RStudio environment -2012.09.0 / Build 351)

and the correlation matrices were visualised using the 'corrplot' function [START_REF] Wei | R package "corrplot[END_REF]. Blue and red colours in correlation matrices indicate positive and negative correlation, respectively. The shade of the colour and the circle diameter refers to the amplitude of the correlation.

  peak 4+ peak 5+ peak 6+ peak 7+ peak 8+ peak 9)+2*(peak 10+ peak 11) De-acetylation = peaks 4+5 De-acetylation of dimers = (peak 4+ peak 5)/(peak4 + peak5 + peak6 + peak7 + peak8 + peak9) DL-EP products = M2 = peak 3 DD-EP products = M3 + M3-NH2 = peaks 1+2

Figure 1 .

 1 Figure 1. Schematic view of the PG synthesis pathway in B. subtilis. The first step is the synthesis of

Figure 2 .

 2 Figure 2. Quantification of total PG and soluble PG precursors upon antibiotic treatment Results shown are mean of two independent experiments, standard deviation is presented as error bars. Results are shown for ABC. 2h protocol, DEF. Overnight protocol

Acknowledgments

This project has received funding from the European Research Council (ERC) under the Horizon 2020 research and innovation program (grant agreement No 772178 to R.C.-L.). Research in the Cava lab is supported by the Swedish Research Council (VR), The Knut and Alice Wallenberg Foundation (KAW), The Laboratory of Molecular Infection Medicine Sweden (MIMS) and The Kempe Foundation.

Disclosure and competing interests statement

The authors declare that they have no conflict of interest.

A&D -Growth curves with the high concentration of antibiotic (in orange in Fig. S1C&D). B&E -Corresponding total PG amount expressed as a percentage of the untreated control.

C&F -Precursors quantification expressed as a percentage of the untreated control. Each precursor is presented as the sum of the non-amidated and amidated form (UDP-tri=UDP-M3+UDP-M3-NH2, UDP-penta=UDP-M5+UDP-M5-NH2). De-acetylation of dimers = (peak 4+ peak 5)/(peak4 + peak5 + peak6 + peak7 + peak8 + peak9) C. Correlation between crosslinking (expressed as % of total PG) and de-N-acetylation (expressed as % of dimers) of mutant strains. Each value is a mean of three independent experiments. De-acetylation of dimers = (peak 4+ peak 5)/(peak4 + peak5 + peak6 + peak7 + peak8 + peak9) D. Correlation between crosslinking (expressed as % of total PG) and amidation (expressed as % of total possible amidation) for all antibiotics, for the two protocols. Amidation = % of amidated species vs all possibly amidated species = 100* (peak1 + 0.002*peak3 + 2*peak4 + peak5 + 2*peak6 + peak7 + peak8 + 2*peak9 + 2*peak10 + 3*peak11) / ((peak1 + peak2 + 0.002*peak3) + 2*(peak4 + peak5 + peak6 + peak7 + peak8 + peak9) + 3*(peak10 + peak11)) E. Correlation between crosslinking (expressed as % of total PG) and amidation of dimers (expressed as % of total possible amidation of dimers) for all antibiotics, for the ON protocol. Amidation Dimers = 100* (2*peak4 + peak5 + 2*peak6 + peak7 + peak8 + 2*peak9) / (2*(peak4 + peak5 + peak6 + peak7 + peak8 + peak9))

D, E. Correlation between crosslinks and M3 monomers (D) and between crosslink and M3-NH2

monomers (E) expressed as a percentage of total PG features (mean of two independent experiments). F, G. Relations between the monomers and their respective variations, for penicillin-and vancomycintreated cells (F) and for dacA mutant (G).