Adrien Jacquot

Chizuko Sakamoto

Angelina Razafitianamarahavo

Céline Caillet

Jenny Merlin

Ahmad Fahs

Jean-Marc M Ghigo

Jérôme F L Duval

Christophe Beloin

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

A Razafitianamaharavo

The dynamics and pH-dependence of Ag43 adhesins' self-association probed by atomic force spectroscopy

Keywords: Bacterial adhesins, Antigen 43, unwinding, molecular sliding, AFM

Introduction

Bacterial adhesion onto abiotic surfaces and host tissues, as well as bacteria-bacteria interactions are key processes for colonization, biofilm formation and further infection. These different interactions are commonly mediated by surface appendages exposed at the bacterial cell surface. 1,2 Large heteropolymeric appendages such as fimbriae are generally used by the bacteria to interact with surfaces over large separation distances, while shorter adhesins such as self-associating autotransporters (SAATs) adhesins transported by type V secretion are capable of promoting intimate interactions between bacteria. 3,4 Prototypical members of this family of SAAT adhesins are AidA, an adhesin originally characterized in an E. coli O126:H27 strain isolated from a pediatric patient with diarrhea, 5 TibA, first found in the ETEC O78:H11 strain H10407, 6 and the Antigen 43 adhesin (Ag43), one of the most abundant outer membrane proteins in E. coli. 7,[START_REF] Owen | Electroimmunochemical analysis of membrane proteins[END_REF] Like other AidA-I-type adhesin autotransporters, E. coli Antigen 43 is a short and rigid, monomeric adhesin composed of a Cterminal integral outer membrane β domain through which is exported a N-terminal passenger α domain (Ag43α). [START_REF] Wells | [END_REF]10 After its autocatalytic cleavage, Ag43α (499 aa) remains loosely anchored to Ag43 β domain and forms a β-strand helix that protrudes 10 nm from the cell surface. [11][12][13][START_REF] Heras | Proc Natl Acad Sci[END_REF] Like other AidA-I-type adhesin autotransporters, this adhesin is able to promote cell-cell interactions, and subsequent bacterial autoaggregation 11,[START_REF] Hasman | [END_REF] and biofilm formation 3,[16][17][START_REF] Henderson | Mechanisms in the Pathogenesis of Enteric Diseases[END_REF] through homomeric recognition of its α-domain potentially folded as a β helix. Some Ag43 variants have been further shown to facilitate the persistence of uropathogenic E. coli within the bladder, [START_REF] Ulett | [END_REF]20 to promote uptake of-and resistance to-antimicrobial agents and protection of E. coli against polymorphonuclear neutrophils. 11,21 In 2011, the structure of the α-domain of the AT Hap protein from Haemophilus influenzae was resolved, which opened a route for elucidating the relationship between structure and homophilic interaction mechanisms of the SAATs. 22 In the Hap protein, homophilic recognition is promoted by the interaction in trans configuration of the α-domain along a 7turn right-handed β-helix, with ~19 residues in each turn. More recently, Heras et al. resolved the crystallographic structure of the Ag43a variant α-domain of uropathogenic E. coli strain CFT073. [START_REF] Heras | Proc Natl Acad Sci[END_REF] As previously predicted, 11 they demonstrated that the functional 499 amino-acids α-domain of this SAAT protein is a 3 stranded β-helix structure comprising 23 complete turns with an average of 20 amino-acids per turn, each turn being composed of three faces marked by the arrows depicted in Figure S2, thus forming a triangular structure. Interestingly, Ag43aα displays a twisted L-shaped β-helical structure that plays a major role in the selfassociation and cell aggregation via a molecular "Velcro-type" mechanism where the dimerization of the α-domain is ensured by multiple hydrogen bonds between the two twisted beta-helical structure coil around each other in a trans/head-to-tail configuration. [START_REF] Heras | Proc Natl Acad Sci[END_REF] A modeling of the α-domain of E. coli K-12 Ag43 that share 81.7% identity with that of UPEC CFT073 Ag43a, revealed that this domain also probably displays a twisted L-shaped β-helical structure (Figure S2). However, little is known on the self-association of beta-helical proteins, in particular its dynamics, its dependence on medium composition including pH, and the way it is connected to the intrinsic mechanical and structural properties of the interacting proteins domains. These key issues remain difficult to address at the nanoscale from crystallography or X-ray scattering techniques (SAXS, WAXS) and macroscopic observations of self-aggregation of bacterial cells are useless in this perspective too.

Since the last decade, derivative techniques of atomic force microscopy (AFM) were developed for probing the physico-chemical properties of biomolecules. [23][24][25][26] Techniques such as Dynamic Force Spectroscopy (DFS) and Single-Molecule Force Spectroscopy (SMFS) now allow access to molecular dynamic processes, chemical or surface reactivity, conformational properties, and interactions between single molecules. 27,[START_REF] Alsteens | Proc. Natl. Acad. Sci[END_REF] For the sake of example, Single-Molecule Force Spectroscopy (SMFS) was recently used to unravel and explain the binding mechanism of BCAM0224 auto-transporter adhesins involved in respiratory infections and secreted by Burkholderia cenocepacia. [START_REF] El-Kirat-Chatel | [END_REF] The authors evidenced that the BCAM0224 adhesin behaves like a nanospring when subjected to external shear forces and can form both homomeric and heteromeric interactions that enhance adhesion to immune cells with corresponding forces as large as 120 pN. [START_REF] El-Kirat-Chatel | [END_REF] Using a similar technique, the nanomechanical properties of LapA adhesin expressed by Pseudomonas fluorescens, or type 1 fimbriae attached to mannose were investigated under various pH, ionic strength and shear force conditions. 30,31 In this work, we report a systematic investigation of the impact of solution pH and externally applied shear force on the dynamics of E. coli K-12 Ag43 α-domain (Ag43α) unfolding and molecular self-association. The conditions tested in this work mimic those met by E. coli cells in their natural environment. Our experiments are carried out using functionalized AFM probes specific to Ag43α interacting with either model surfaces functionalized by Ag43 adhesins or with Ag43-decorated E. coli K-12 cells. Retraction force curves were recorded at different pH values ranging from 3 to 9 and were analyzed on the basis of the Worm Like Chain (WLC) model in order to quantify the impact of pH on the conformation properties of Ag43 structure. Dynamic force spectroscopy (DFS) measurements were further performed to investigate the dynamics of Ag43 self-association and unfolding under a wide range of pH and pulling rate conditions. Results obtained from DFS and SMFS experiments were finally supported by macroscopic bacterial auto-aggregation assays. We show that self-association forces between Ag43 α-domains are mainly driven by trans interactions of α-domains as strong as 100-250 pN, and we suggest the further contributions of possible cis interactions between these domains at sufficiently large values of pH and elongation rate. In addition, the existence of a tetrameric structure composed of Ag43 dimers is hypothesized from inspection of molecular stretching data collected by AFM. Such structure results from the coexistence of trans-and cis-interactions, and its existence is mediated by the lifetime of these interactions and by the solution pH conditions. Our results bring novel information on the mechanical properties of β-helical proteins and on the dynamics of their homomeric interactions.

Results

AFM nanogold-mediated analysis of cells expressing or not Ag43 adhesins.

The surface properties (morphology and adhesion) of E. coli K-12 cells constitutively producing or not Ag43 (E2498 and E2152, respectively) were investigated by AFM in PeakForce Mapping ™ mode using a functionalized His-Ag43α coated tip where the histidine tag was fused at the N-terminal part of Ag43 α-domain. The obtained adhesion maps collected with the functionalized AFM-tip (Figure S1) revealed that the only E2498 strain exhibited a moderately adhesive bacterial cell wall surface. These results confirmed the presence of Ag43 adhesins over the cell wall of the E2498 strain and they further highlight the specific self-recognition of Ag43 located at the bacterial surface and on the AFM tip. To further localize the Ag43 adhesins by virtue of Ag43α-Ag43α homomeric interactions, we imaged bacterial cells constitutively producing Ag43 or not after incubation with nanogold particles grafted with His-Ag43α where the histidine tag is fused at the N-terminal part of Ag43 α-domain. When experiments were performed with cells E2498 constitutively expressing Ag43, we observed the presence of spherical structures 20 to 200 nm in diameter over and around the bacterial cells (Figure 1a, white arrows). NTA-nanogold particles are 5 nm in diameter before functionalization and the estimated size of Ag43α domain is 10 nm.

Large structures (> 20 nm) are then supposedly composed of 10-25 nm diameter substructures interacting following homomeric recognition of Ag43α. These large structures did not correspond to aggregated nanogold particles and their observation depended on the presence of Ag43 on the E. coli cell surface since they were not detected for the strain E2152 deleted for the agn43 gene (Figure 1b). These results support that the presence of the aforementioned structures results from homomeric interactions taking place between Ag43α carried by nanogold and Ag43α located at the E. coli cell surface. The presence of structures with characteristic dimensions of ca 200 nm suggested that more than two molecules of Ag43α may interact in order to form such large complexes.

Analysis of Ag43α self-association by dynamic force spectroscopy

The dynamics of the adhesin conformational structure during self-recognition process is an important issue to address for understanding the auto-aggregation mechanism and the nature of the underlying protein-protein interactions. Single Molecule Force Spectroscopy (SMFS) and Dynamic Force Spectroscopy (DFS) are very relevant tools in that respect because they enable the analysis of the protein conformational changes and dynamics thereof during protein-protein interactions on living cell surfaces. [START_REF] Alsteens | Proc. Natl. Acad. Sci[END_REF]36 Therefore, combined SMFS and DFS experiments were performed here in order to (i) evaluate the interaction forces taking place between two Ag43 α-domains, (ii) investigate the dynamics of unfolding, and (iii) address the impact of pH and external shear forces on the protein conformational and structural features.

First, both gold surfaces and gold AFM-tips were functionalized by the histidine-tagged Ag43α after thiol-NTA grafting (N-ter fusion). The typical force curves corresponding to the interaction between Ag43α-functionalised gold surface and Ag43α-functionalised gold AFMtip are reported in Figure 2 at pH 7.4 for 100 and 3200 nm/s pulling rate conditions and in Table 2 for intermediate values of pulling rates. The corresponding statistic distribution of the number of rupture events observed while retracting the modified-tip from the Ag43-modified surface are further displayed in Figure 2. The specificity of the functionalized AFM tips to Ag43α was certified from the absence of interaction between nude gold surfaces and the modified tips (data not shown). Figure 2 displays qualitatively similar retraction force curves for 100 and 3200 nm/s pulling rates with the presence of multiple adhesive events.

While such event could be detected for tip to surface separation distance as large as 150 nm at 3200 nm/s pulling rate (Figure 2b and Figure S4), Ag43α uncoiling was observed over significantly shorter separation distances (40-70 nm) at 100 nm/s (Figure 2a and Figure S4).

Overall, the unfolding distance appears 5 to 15 times larger than the putative size of the folded adhesin of 10 nm dimension, which reflects a significant stretchability of this protein.

The structured part of the α-domain is supposedly folding as 23 complete coils with each spiral pitch containing an average of 20 amino acids ( 14 and Figure S2). Since the minimal force required to totally and concomitantly unfold two β helical structures is above 400 pN, [START_REF] Junker | Proc. Natl. Acad. Sci[END_REF]38 this means, theoretically, that only one β helical structure can be totally unfolded and thus a maximum of 23 rupture events should be observed by force spectroscopy measurements (note that the typical range of measured rupture force for self-associated Ag43α is ca. 100-200 pN, cf. Table 2). The average number of ruptures per force curve we found decreased from 7 ± 3 to 3 ± 2 when the pulling rate was increased from 100 to 3200 nm/s (Figures 2c and2d).

These results evidenced a sequential unfolding of the interacting α-domains but with a maximum of only 5 to 10 ruptures. Basically, this suggests that the interaction between two α-domains breaks up before complete unfolding of the structures is achieved. Additionally, this result indicates that Ag43α is folded in several domains composed of multiple turns, and consequently the unfolding is sequential and takes place domain after domain and not turn after turn.

The retraction force curves depicted in Figure 2 were further analyzed according to the Worm like Chain (WLC) model (eq 1) according to the methodology we detailed elsewhere. 39 Briefly, this theoretical approach allowed the evaluation of several key parameters pertaining to the structural properties of Ag43α and to their homomeric interactions:

-the persistence length l p reflecting the intrinsic flexibility of the protein; -the contour length L c defined as the length of the fully extended protein; -the rupture force between two proteins; -the number of monomers, denoted as N, for the stretched system.

The statistic distribution of the last rupture force magnitude detected in the Force volume Image (FVI) and the number of monomers extracted from the retraction force curves are collected in Figure 3 for pulling rates of 100 and 3200 nm/s at pH 7.4. The rupture forces measured between two adhesins were not significantly modified by the pulling rate, with values of 138 ± 35 pN and 141 ± 60 pN at the two rate values adopted here. These values of rupture force measured onto model Ag43α-gold coated surfaces are comparable with those measured on living cells (Figure S1a-3). Moreover, the statistical analysis of N, corresponding to the number of monomers within the Ag43-Ag43 system, is bimodal, which suggests that at least 2 sub-structures were stretched during the retraction of the AFM-tip from the functionalized gold surface. We hereafter refer to N 1 and N 2 (with N 1 < N 2 ) these two values taken by N, as illustrated in Figure 3c,d. We obtained N 1 = 422 ± 224 and N 2 = 1465 ± 272 at 100 nm/s, and N 1 = 418 ± 156 and N 2 = 1938 ± 711 at 3200 nm/s (see Table 2). .In order to address the nature of the two types of substructures probed by force spectroscopy at 100 and 3200 nm/s and pH 7.4, we plotted in Figure S5 the persistent length derived from WLC analysis as a function of the corresponding number of monomers N. The data clearly evidence the presence of two distinct clouds of points corresponding to the two searched substructures. In details, one may discriminate a substructure (1000 ≤ N ≤ 2500) with a persistence length l p lower or equal to 0.15nm, and a second type of substructure (100 ≤N ≤1000) with a l p satisfying l p >0.15 nm. The former corresponds to a weak stretching of C-C bonds and the latter to the stretching of amino acids. Indeed, the length of a C-C bond is in the range 0.12-0.15 nm, which comes to state that all persistence lengths larger than this value must necessarily correspond to amino acids. Figure 3 shows that the bimodal distribution in terms of number of monomers does not depend on the pulling rate value. Additional experiments further show that the aforementioned bimodal distribution is not significantly affected with changing the solution pH (Figure S6 and Table 3).

Interestingly, a statistical switch in the respective weight distribution of the two substructures probed by force spectroscopy is observed with increasing the pulling rate from 100 to 3200 nm/s, as judged from the comparison between data collected in Figure 3c andFigure 3d. In details, the probability to observe the stretching of amino acids (N 1 ~ 500) was about 35% at 100 nm/s and increased to 57% with increasing pulling rate to 3200 nm/s. For an intermediate value of the pulling rate (400 nm/s), an iso-distribution of the two substructures is even reached regardless of the pH conditions (Figure S6). These results highlight the intrinsic dependence of the extent of stretching of the C-C bonds and that of the amino acids on the applied pulling rate. To put it a nutshell, a slow elongation dynamics (low pulling rate values) predominantly favors the stretching of large assemblies of amino acids (Figure S5) whereas faster elongation dynamics (probed at higher pulling rates) goes in pair with a stretching of smaller C-C bonds. The number of monomers corresponding to the amino acid structures is in the range 100 to 1000 (Figure S5). However, the complete stretching of two ß helical structures of Ag43 should lead to a number of monomers equal to 499×2 amino acids [REFERENCE]. Obviously then, the rupture forces in our experiments (≤216 pN, see Table 3) are not large enough to totally unfold simultaneously two ß helical structures, which is in agreement with the literature (250 pN at least are required to unfold two ß helical structures) (REFS).

Effect of pH and pulling rate conditions on the dynamics of Ag43 unfolding

In order to further investigate the changes in Ag43 conformation and structure during selfassociation of the proteins, force spectroscopy experiments were carried out at different pulling rates in the range 100-3200 nm/s with varying the solution pH (5.2, 6.1, 7.4, 8.1 and 9.2). We emphasized that the pulling rate set in the force spectroscopy experiments basically fixes the time window where the self-association modes of the Ag43α domains can be observed. Stated differently, with increasing the pulling rate, one will probe the fastest Ag43α association mode and, conversely, the lower the pulling rate, the more efficient will be the observation of the slower association mechanisms. In fine, varying the pulling rate allows us to modulate the measurement timescale and thus to vary the so-called Deborah number defined here as the ratio between Ag43α association timescale and measurement timescale.

Force curves similar to those given in Figure 2 were then recorded as a function of pH and analyzed on the basis of the WLC model. A marked dependence on pH and pulling rate was detected for the maximal molecular extension denoted as L max (corresponding to the maximal rupture distance of self-associated Ag43, Figures 4a and4b) and the magnitude of the last rupture force was further found to be strongly modulated by pH too (Figure 4c and4d). In details, L max varied with pH according to a sigmoid-like dependence and reached plateau values at sufficiently low and large pH values. For the lowest pulling rate tested in this work (100 nm/s, Figure 4a), the plateau value under acid conditions was about 40 nm (corresponding pH range: 5 to 7), while the plateau reached at pH values larger than 9 was 80 nm. For higher pulling rates (3200 nm/s, Figure 4b), the same trend was observed, albeit with slightly higher plateau values reached by L max for pH < 7 (50-65 nm) and pH > 9 (100-150 nm). In addition, the magnitude of the last detected rupture force depended on pH according to a bell-shaped curve regardless of the pulling rate value adopted in the experiments. At low pulling rates, a shallow maximum is obtained around neutral pH with a value of about 120-150 pN (Figure 4c). For higher pulling rates, the maximum was more pronounced with a value of 120-150 pN reached at neutral pH, similarly to the situation at lower elongation rates (Figure 4d). For pH higher than 8, weak ruptures forces are in line with a significant extension of the structures. For pH lower than 6, rupture forces are weak and they correspond to a minimal extension of the structures (Figures 4a and4b).

We investigated in more details the dependence of the rupture forces between Ag43 proteins on pulling rate (Figure 5). Over the whole range of pH conditions tested, we observed a belllike dependence of the rupture force on pulling rate. Under the three pH conditions reported in Figure 5, the rupture forces gradually increased from 80-120 pN to a maximum value in the range 150-220 pN at a critical pulling rate in the range of 400-800 nm/s, and finally it somewhat decreased upon further increase of the pulling rate. This surprising feature was detected for pH values in the range 3 to 9, even though the bell-like dependence seems to be less pronounced at pH 9.2. A three-dimensional representation of the dependence of the rupture forces on pH and pulling rate, as given in Figure S7, can be fitted by a 3D-Gaussian form in order to estimate the pH and pulling rate conditions that are most favorable for strong Ag43 association. We found that these conditions correspond to a pH of about 7.6 and a pulling rate of 473 nm/s. The increase of the binding force between the self-associated adhesins with increasing the pulling rate is fully in line with a "catch-bond"-like mechanism where the receptor-ligand interaction is enhanced with increasing tensile mechanical force.

Macroscopic signature of Ag43-Ag43 assembly and pH dependence.

In order to evaluate the relevance -at a macroscopic scale -of the observed impact of solution pH on the self-association of Ag43 adhesins, we performed aggregation measurements over the pH range 3 to 9 on bacterial cells (E. coli -E2498) constitutively expressing or not the Ag43 adhesin. We report in Figure 6 the optical density ratio OD/OD 0 that reflects the fraction of non-aggregated (and non-settled) bacterial cells in solution. After 60 minutes, the OD/OD 0 ratio followed a marked parabola-shaped trend versus pH, with a pronounced minimum at pH ~ 5. After 120 and 180 minutes, the parabolic trend became strongly attenuated but a minimum in the OD/OD 0 ratio remains still clearly distinguishable at pH 5. In line with previous work from our group, 33 we verified that the OD/OD 0 ratio for the control strain (E2152) lacking the Ag43 proteins at the surface is about 1 over the time period tested in Figure 6 regardless of the pH conditions The minimum in OD/OD 0 ratio observed for the E2498 strain should thus correspond to an optimum in aggregation as mediated by the selfassociation of Ag43 adhesins located on the surface of the interacting cells. In agreement with Figure 6, self-association of Ag43 (and therefore bacterial aggregation) is expected to be most significant at pH values close to the theoretical isoelectric point pI=4.83 of the Ag43 αdomain (calculated using http://web.expasy.org/compute_pi/). A mechanistic understanding of these results requires a molecular analysis of the Ag43 adhesins self-association modes as revealed by force spectroscopy experiments (see Discussion section). It is stressed that macroscopic measurements of the electrohydrodynamic features of bacterial cell walls 40 are not informative in that respect, as thoroughly detailed in ESI (Figure S8).

Discussion

The mechanical properties of bacterial adhesins play essential roles in several biophysical and interfacial processes, such as adhesion to abiotic or host cell surfaces, 41 biofilm formation, 16,[42][43][44][45] auto-aggregation and pathogenicity. [START_REF] Ulett | [END_REF]46,47 The combination of single-molecule force (SMFS) and dynamic force spectroscopies (DFS) makes it possible to address the molecular mechanisms underpinning how pathogens surface proteins contribute to host cell recognition, adhesion and infection and how processes including catch bond effect, 48 unfolding, 49 unzipping, [START_REF] Muller | Proc. Natl. Acad. Sci[END_REF] and nanospring behavior 58 come into play. Our results provided the first dynamic and biomechanical investigation of Ag43 self-association modes under different pH and applied shear force conditions.

Mechanical behavior of Ag43 self-association

Force spectroscopy measurements are in line with the previously reported ß-helix structure of the Ag43α domain [START_REF] Heras | Proc Natl Acad Sci[END_REF] essentially because our data support that Ag43α unfolding is sequential and the magnitude of the unfolding forces is in the range of 100-250 pN. 38 The latter well agrees with that given for homophilic association of BCMA0224, a prototypic trimeric autotransporter from B. cenocepacia, and HBHA from M. tuberculosis. [START_REF] El-Kirat-Chatel | [END_REF][START_REF] Verbelen | [END_REF] The here-reported unfolding forces are also within the same range of those measured for the unfolding of single adhesins such as LapA from P. fluorescens and FHA from H. influenza. 49,52 Interestingly, homophilic interactions of adhesins appear stronger (4-to 8-fold) than the interactions between fimbrial adhesins and their corresponding sugar ligands. Forces between 30 to 50 pN have been reported for such fimbrial adhesins/sugar interactions. 31,[53][54][55] A sequential unfolding of non-fimbrial adhesins reflects the presence of several structural domains such as those recently reported for the FHA protein from H. influenzae. 49 In the case of Ag43α, depending on the adopted pulling rate value, we demonstrated the presence of 3 to 10 structural domains that correspond to the number of rupture events detected in the force curves. Heras et al demonstrated that the Ag43α domain of Ag43a in uropathogenic E. coli strain CFT073 exhibits a L-shaped structure consisting of a 13-rung parallel β-helix domain, followed by three rungs that bend the protein by about 110° and a C-terminal seven-rung parallel β-helix domain. [START_REF] Heras | Proc Natl Acad Sci[END_REF] In this structure, 3 domains can be identified and could be further visualized from structure modeling of Ag43α in E. coli K-12 (see Figure S2): the first of these domains consists of 13 turns of 292 amino acids (green ribbon), the second contains 3 turns of 69 amino acids (red ribbon) and the last domain is composed of 7 turns in a sequence of 137 amino acids (yellow ribbon) (see Figure S2). Our AFM data therefore suggest that these 3 domains contain sub-structures that are composed of several turns and that constitute independent structural segments.

The analysis of force experiments performed with increasing pulling rate revealed a catchbond-like behavior of Ag43α. Indeed, a rise in the rupture forces from 120 up to 220 pN has been measured at pH 7.4 when the pulling rate was increased from 100 up to 1000 nm/s (Figure 4 and Figure 5). In literature, "catch bond" has already been described for fimbrial adhesins (type 1, F1C and P) with an increase of the force from 30 pN up to 70 pN for pulling rate values in the range of 100 to 10000 nm/s. [START_REF] Axner | Bacterial Adhesion: Chemistry, Biology and Physics[END_REF] Catch bonds may correspond to several mechanisms allowing the bond lifetime to increase with mechanical stress until a critical shear force condition is reached. Such mechanisms confer bacteria the ability to roll under low shear forces and to strongly adhere at higher shear forces, as in e.g. urinary flows. In our experiments, the 80% increase in the magnitude of the rupture forces between two Ag43α with increasing pulling rate from 100 to 1000 nm/s is well in agreement with that measured for the P pili from uropathogenic E. coli. [START_REF] Andersson | [END_REF]58 This result basically indicates that autoaggregation and adhesion of the bacteria expressing Ag43 onto Ag43-coated surfaces would be favored with increasing flow rates. Even though the molecular mechanism remains quite unclear, we think that the strong pulling rate-dependence of the adhesion force between Ag43α is a signature of catch-bond effects. This effect observed for the self-association of Ag43α domains is maximum at pH 7.4 and is strongly reduced under both low and high pH conditions (Figure S7,Figures 4c,d). Interestingly, this pH value where the rupture force is maximum roughly identifies with that where the curve maximal extension L max versus pH exhibits an inflection point (Figures 4a,b). Overall, these results firmly demonstrate that the auto-assembly of Ag43 proteins is of a dynamic nature and that it depends on solution pH.

Is the formation of self-associated Ag43 adhesins mediated by trans-and/or cisinteractions?

Heras et al. have clearly identified 2 loop regions (depicted in Figure S2 in pink and blue colors) corresponding to the so-called 1 and 1' interfaces recognized as the molecular promoters of self-recognition mechanism. [START_REF] Heras | Proc Natl Acad Sci[END_REF] These authors further stated that self-association of two Ag43α domains takes place through the interfaces 1 and 1' in trans head-to-tail interaction acting like a molecular Velcro (Figure 8). Considering such an interaction configuration via the interfaces 1 and 1', we estimated that the maximal extension before rupture would correspond to the unfolding of the 22 amino acids sequence located before interface 1 and/or the 66 amino acids sequence located between interfaces 1 and 1', as schematized in Figure 8. The concomitant stretching of two 22 amino acids sequences located on both sides of interfaces 1 and 1' is also possible. Such molecular conformations should lead to maximal molecular extensions of 7, 14, 24 and 38 nm (Figure 8). The first and third values stem from the unfolding of only one Ag43 protein. The concomitant unfolding of the two 22 amino acid sequences would lead to 7+7=14 nm (in the case where the sequence between 1 and 1' interfaces is not unfolded) or to a maximal extension of 7+22+7=38 nm (in the case where the entire sequence along the stem of the L-shaped structure is unfolded). The unfolding mechanisms schemed in Figure 8 result in maximal molecular extensions that are in line with those collected in Figure 4 at sufficiently low pH values and low pulling rates (typically 100 nm/s, Figure 4a). This strongly suggests the occurrence of trans-association of Ag43α domains under such pH and pulling rate conditions.

For pH values higher than 7 and pulling rates larger than say 400 nm/s, we systemically observed a maximal extension up to 150 nm for the interacting Ag43α domains (Figure 4).

One could argue that such large extension could be achieved from the cascade disruption of several trans-associated Ag43, in a manner similar to that observed for the autotransporter SAAT adhesin Hap from H. influenzae. 22 However, a cascade stacking of Ag43 is not possible because, unlike Hap which is a straight solenoid, Ag43 exhibits a bend that necessarily leads to strong steric hindering of the access to the interfaces 1 and 1'. Therefore, in order to explain values as large as 150 nm for the maximal extension of Ag43α domains (Figure 4), it is mandatory to argue another possible configuration for Ag43α homophilic association, i.e. a configuration other than that previously reported from crystallography analyses. [START_REF] Heras | Proc Natl Acad Sci[END_REF] We suggest that Ag43α interactions via the interfaces 1 and 1' following transassociation predominate at low to intermediate pulling rates and/or low pH (see discussion above) with as a result a moderate extension of the structure (ca 38 nm at the largest, Figure 4). Since molecular extension above 38 nm cannot result from stretching of Ag43 associated according to trans-interactions, another type of interactions explaining the large molecular extensions observed at large pulling rate (Figure 4b) and to some extent at large pH values should exist. As further discussed in this section, we hypothesize that such interaction between Ag43α is faster and weaker than the trans-interaction detected under low pulling rate conditions. In literature, it is reported that some proteins (Als5p adhesin from C. albicans 59,60 , Nectin proteins 61 ) may associate side-by-side in cis-configuration. If adopting such a configuration for Ag43α dimers where the bends of the L-shaped proteins are in opposite direction (Figure 9a) and further considering the sole unfolding of one out of the two interacting molecules, it can be estimated that the maximal extension expected should be around 105 nm (corresponding to 292 amino-acids). In the case where the bends of the Lshaped proteins are in the same direction (Figure 9b), the maximal extension is around 160 nm (corresponding to 499 amino-acids). Both values 105 and 160 nm obtained from such unfolding mechanisms very well correspond to the maximal extensions measured by AFM at sufficiently large pH values and/or large pulling rates (Figure 4 and Figure S7). To the best of our knowledge, cis-interactions have never been invoked for autotransporter adhesins for which self-assemblies are deemed to take place only from trans-interactions. [START_REF] Heras | Proc Natl Acad Sci[END_REF]22 Our results do support the occurrence of such interaction configuration. With regard to the continuous increase in maximal molecular extension with increasing pH and/or pulling rate (Figure S7b), it can be argued that trans-interactions are predominant at pH < 7 and pulling rates lower than 400 nm/s and that they lead to moderate maximal extension of the structure. Conversely, cisinteractions are probably the major mode of protein association at pH > 7 and at pulling rates larger than 400 nm/s, and they are in line with molecular extensions that are significantly larger than those obtained according to trans-interactions mode. The pH and pulling rate regimes where cis-and trans-interactions are most significant are schematized in blue and red-orange color mapping in Figure S7b. The buffer space defined by the green area in Figure S7b can be associated with the overlap between trans-and cis-interaction zones of influence. This coexistence of cis-and trans-interactions as well as the values obtained for N 1 (> 400 amino acids) suggests a possible multimerization where two or more Ag43α molecules are involved within the molecular assembly and subsequent unfolding upon retraction of the modified tip. For the sake of example, a tetrameric structure of Ag43α is possible through the trans-interaction between two adhesins dimers associated via cis-interactions (Figure 10). Such molecular configuration is also supported by nanogold particles-mediated observations (Figure 1a) that revealed the occurrence of large structures of ca. 200 nm, which probably corresponds to large complexes consisting of more than two Ag43α molecules.

Building mechanism and unfolding of Ag43α tetrameric structure

Multimerization of adhesins has been largely documented in literature and particularly for autotransporter adhesins such as Hap from Haemophilus influenzae, AidA-I from Escherichia coli and YapC from Yersinia pestis. 62,63 In addition, the formation of homo-tetramers starting from homo-dimers has been reported for the repair protein MutS from Escherichia coli 64 and also for the autotransporter adhesin BadA from Bartonella henselae. 65,[START_REF] Hartmann | Proc. Natl. Acad. Sci[END_REF] The authors stated the existence of equilibrium between dimers and tetramers and the necessity of full-length dimers interactions to achieve the tetramer formation. 64 The analysis of the modulation of the number of monomers and the maximal extension L max (Figure 3 and Figure 4, respectively) evidenced that molecular unfolding is quasi-completed (i.e. L max ~160nm) under restricted range of pH and pulling rates (pH~9 and pulling rate ν > 1600 nm/s, cf. Figure S7b) where cis interactions are predominant (Figure 9). Under such extreme ranges of pH and pulling rates, trans-interactions cannot take place since, in such a configuration, the only stem of the L-shaped structure of Ag43 can be unfolded leading to L max ~38 nm at most, in contrast to experimental data. In order to explain the continuous increase of L max from ~30 nm to ~160 nm with increasing pH from 3 to 9 and/or pulling rate from 100 nm/s to 3200 nm/s, configurations intermediate between those corresponding to trans and cis interactions or combinations thereof must be envisaged. Such combinations of trans and cis interactions can lead to multimers-assemblies for Ag43, as reported for e.g. MutS and BadA adhesins. [64][65][START_REF] Hartmann | Proc. Natl. Acad. Sci[END_REF] In this situation, Ag43 dimers are formed by cis-interactions with the bends of the L-shaped structures in the opposite direction before interacting via interfaces 1 and 1' in a transconformation to finally achieve a tetrameric structure (Figure 10). The occurrence of such a structure is in line with the presence of free Ag43α molecules in excess in the solution used for surface functionalization. These free molecules can indeed form dimers with the Ag43α molecules attached to the gold-surface or to the AFM tip by virtue of rapid cis interaction. As discussed below, stretching of the above tetrameric structure can lead to molecular extensions between 30 nm and 210 nm, in line with extensions data collected in Figure 4.

The analysis of the force curves collected in the pH range 4 to 9 evidenced a strong correlation between the pH-dependent maximal unfolding of the self-associated adhesins and the corresponding rupture force (Figure 4). Indeed, a systematic increase of the maximal molecular unwinding was observed with increasing pH from ~7 to 9, i.e. in the pH range where a maximum in the rupture forces was detected. The theoretical isoelectric points of the interfaces 1 and 1' (involved in the sole trans interactions) can be estimated from a proper account of the amino acids constituting these interaction regions between two sets of residues in each monomer (see Figure S2), with, as result, a theoretical value of 7.10. This value is close to that obtained from AFM rupture force measurements. It thus strongly suggests that this maximum in rupture force between the Ag43 adhesins is predominantly governed by the trans-interactions and to a lower extent by the cis interactions (that come into play at larger pH values). The existence of two plateau regimes for the maximal molecular extension, i.e. at sufficiently low and large pH values, may be further explained as follows. The low molecular extension value observed under acidic conditions could originate from the favored formation of H-bonds between the β-helices. This would necessarily increase the intra-molecular cohesion forces within the β-helices interacting in trans-conformation and thus reduce the stretchability of the formed dimer. The high molecular extension observed under alkaline conditions could stem from an increase of the electrostatic charges located in the β-helices.

The subsequent increase in the repulsive interactions could enhance the dimer flexibility and thus increase its stretchability as compared to the low pH situation. Finally, the increase in the rupture forces with pH could be a signature of the abundance of self-associated β-helices or a signature of the switch from trans-to cis-interactions for a given dimer.

The results obtained for the maximal molecular extensions and the coexistence of cis-and trans-interactions support the existence of possible multimerization of Ag43 adhesins. The formation of a tetrameric structure composed of only Ag43α adhesins and its unfolding are possible only on the premise that cis-interactions are weaker with a shorter life-time (i.e. accessible at large pulling rate or fast pulling dynamics) than trans-interactions that are stronger but with a longer life-time (slow dynamics). In view of the respective magnitude of the cis and trans interactions, the stretching of the tetramer leads to the unfolding of the dimers till breaking of the trans-interaction is reached. In that situation, molecular extensions between 30 nm and 210 nm may be achieved, as illustrated in Figure 10. Interestingly, our results suggest that Ag43α may have the properties of promoting -by cis-interaction-the gathering of two or more Ag43 molecules within the E. coli membrane (see Figure 7) favoring subsequent tetramers or higher-order oligomers formation between Ag43 molecules from different bacteria. This is derived from the observation of large nanogold assemblies (~ 200 nm) on cell membrane surface (Figure 1a). The presence of such multimeric structures promotes efficient aggregation of bacteria.

Our study illustrates how molecular effects can be correlated to macro-scale observations such as aggregation of bacteria constitutively producing Ag43 (Figure 6). The maximum in the interaction forces measured between Ag43 adhesins by SMFS is at pH close to 7, whereas we observed a faster bacterial aggregation at pH 5 as previously described in the literature (Figure 6). 11,[START_REF] Sherlock | [END_REF]68 This discrepancy could originate from the difference in processes probed at the nanoscale by SMFS and at the macroscopic scale in bacteria aggregation assays. The number of Ag43 molecules involved in self-association process in both situations is probably different since SMFS measurements are related to the detection of single molecules interactions while aggregation of bacteria may be the result of multiple interactions between Ag43α present at the cell surfaces of several E. coli cells. In the latter situation, pH-dependent interactions other than those stemming from the sole Ag43 surely mediate the rate of approach between the cells whose surface is the location of numerous other types of charged ionogenic groups.

Conclusions

In this study, the dynamics and pH-dependence of homomeric interactions between Ag43 adhesins is investigated, recalling that this protein is involved in biofilm formation and bacterial persistence mechanism. Our results support that Ag43 self-association is driven by trans-interactions as originally proposed by Heras et al. In addition, we evidenced the existence of cis-interactions that lead to the formation of Ag43 dimers and tetramers. Results further strongly suggest a coexistence of such cis-and trans-interactions in vivo. Both interactions types are expected to mediate the dynamics of Ag43 complexes formation at the surface of E. coli and to strengthen/enhance self-recognition interactions between bacteria.

Finally, force spectroscopy experiments highlighted that Ag43 molecules have a high stretchability, and that their self-association dynamics is strongly impacted by environmental factors such as pH and magnitude of shear forces.

Materials and Methods

Bacterial strains

The E. coli K-12 strains used in this study are listed in Table 1 where relevant information on their respective construction, antibiotic resistance, genotype and expression of surface appendages can be found. These isogenic strains were constructed from Escherichia coli MG1655 (E. coli genetic stock center CGSC#6300). All strains used in this study contain the gfpmut3 gene linked to the bla ampicillin resistance gene (amp R , 100 µg/ml) that makes them fluorescent, and a deletion of the fliE to fliR genes replaced by the cat chloramphenicol resistance gene (cm R , 25 µg/ml). Our reference strain (E2152) has been additionally deleted for both the fim operon encoding type 1 fimbriae (∆fimA-H::zeo, zeo R 50µg/ml) and the agn43 gene (∆flu::km, km R 50 µg/ml) while strain E2498 constitutively expresses the agn43 gene (kmPcLflu, km R 50 µg/ml) ). 32,43,69

Growth conditions and preparation for AFM experiments

Bacteria were pre-grown overnight at 37°C under agitation (150 rpm) in M63B1 minimal medium supplemented with 0.4% glucose (M63B1glu) and the appropriate antibiotic to select the strain of interest. The following day, fresh M63B1glu medium was inoculated with the overnight culture to an OD 600 of ca 0.05 and cultivated under the same conditions until the biomass reached an OD 600 of 0.5-0.6. Then, 2 mL of the bacterial suspension were removed and placed on a PEI-silicon wafer slide for 30 minutes. The samples were extensively rinsed with PBS solution to remove M63B1glu and placed directly into the AFM closed fluid-cell with 2 mL of PBS solution at the desired pH in the range 3 to 10.

Preparation of His-Tag Ag43α domains

The strain producing 6x-His-Tag Ag43 was constructed using λ-red linear DNA gene inactivation method with help of long primers 70,71 followed by P1vir transduction. The 6x-His-tag was introduced 6 amino-acids after the sequence signal cleavage site of Ag43 in the strain TG using as a template the strain MG1655kmPcLflu constitutively expressing Ag43 to create TG_kmPcLflu-His. 43 We used the following primers to perform the construction: -Ag43.500-5 5-gtttaccggtgcggtggaccggatatttg-3 -His-tag_alpha.L5 5-gtgcatcatcatcatcatcatcacccgggagaaaccgtgaacggcgg-3 -Ag43.ATG+500-3 5-ggcttgacgacctgccagcccttat-3 -His-tag_alpha.L3 5-ggtgatgatgatgatgatgatgcacaacgatgtcagcagccagcac-3.

Purification of the His-tagged α-domain of Ag43 was performed as follows: 2L culture was grown during 16h in LB medium at 37°C. Cells were harvested by low centrifugation and washed in 50 mL 0.9% NaCl, then pelleted and resuspended in 50 mL 0.5 mM Tris-HCl (pH 7.5), 75 mM NaCl. The α-domain of Ag43 was then released from the bacteria cells by heating at 60°C for 30 min and immediately cooled down on ice for 10 min prior to centrifugation (20 min at 6000 rpm -4°C). The detached α-domain contained in the supernatant was then purified on Ni-NTA His bind resin from Novagen and eluted with 500 mM imidazole (20 mM Tris-HCl-pH 8, 0.5M NaCl). The eluted product was immediately dialyzed over 20 mM Tris-HCl-pH 8 and concentrated up to 1.5 mg/mL using Amicon Ultra units (30KDa membrane) from Millipore. Functionality of the purified α-domain of 6x-His-Ag43 used for the functionalization of AFM tips and of Nanogold particles was evaluated by measuring the capacity of the prepared Ag43α solution to inhibit Ag43-mediated autoaggregation of the MG1655∆oxyR Ag43-overproducing strain (Figure S3). Briefly, a 5 mL culture of MG1655∆oxyR was grown in LB during 16h at 37°C, and its optical density adjusted to 3. Cells were harvested, washed once with PBS 1X and finally resuspended either in PBS 1X or in 50-100 µg/mL BSA or purified 6x-His-Ag43. The suspensions were transferred in 5 mL tubes and let to settle at room temperature for 6h and 24h for imaging. An aliquot of aggregated cells was finally transferred on a microscopy slide and cells were visualized using phase-contrast microscopy.

Bacterial Autoaggregation Assay

Aggregation assays were performed as previously described. 72 Bacteria were grown overnight at 37°C under agitation (150 rpm) in M63B1glu minimal medium. The optical density of the culture at 600 nm (OD600) was adjusted to 3.0 after centrifugation and resuspension in PBS solution at the desired pH in the range 3 to 9. 3 mL of each pH-adjusted culture was transferred to 5 mL hemolysis tubes. These tubes were incubated without agitation at room temperature. The OD600 of the upper part of each standing tube culture was determined after 60, 130 and 180 min of settling.

Chemicals and AFM-tip functionalization

Nickel sulfate (NiSO 4 ) and PBS tabs were purchased from Sigma-Aldrich (Sigma Aldrich, Saint-Quentin Fallavier, France), and used as received. NTA-thiols (HS-(CH 2 ) 11 -EG 3 -NTA) and HS-(CH 2 ) 11 -EG 3 -OH thiols were purchased from Prochimia Surfaces (ProChimia Surfaces, Sopot, Poland). PEG linkers were purchased from Hermann Gruber group (Institute of Biophysics, University of Linz, Austria). AFM-tips and gold-coated substrates were chemically modified with Ag43α using NTA-thiol linkers and His tagged Ag43α molecules following a procedure previously described 73 for the detection and uncoiling of Ag43α proteins. Gold tips (NPG-10, Bruker Nano AXS, Palaiseau, France) and gold-coated glass slides were covered by NTA-SAM after overnight incubation in a solution of NTA-thiol 0.1 mM in ethanol. Then, the tips were rinsed extensively with ethanol, dried with nitrogen and incubated 1h in 40 mM NiSO 4 in PBS solution. The tips were rinsed with milli-Q water, incubated with (His) 6 -Ag43α at 0.2 mg/mL in PBS for 2h and then rinsed with PBS and stored at 4°C.

Nanogold particle functionalization and visualization

A solution of 600 µL of 0.5 mg/mL purified His-tagged Ag43α protein diluted in PBS buffer was added to 900 µL of Ni-NTA-Nanogold® (Ref #2082, 5 nm diameter, Nanoprobes, Inc. New York, USA) at 0.5 µM for 60 min at room temperature. Using a 2-fold molar excess of (His) 6 -Ag43α significantly prevents aggregation of gold nanoparticles after functionalization.

Bacterial samples prepared for AFM imaging were incubated 30 minutes with 200 µL of the functionalized nanoparticles solution. Then, the samples were rinsed extensively three times with Milli-Q water to remove excess of nanoparticles and gently dehydrated under nitrogen flow for AFM imaging.

AFM imaging, Single Molecule Force and Dynamic Force Spectroscopy measurements

AFM images were recorded with a FastScan AFM (Bruker AXS, Palaiseau, France) used in Peak Force TM tapping mode. Basically, Peak Force TM tapping mode performs a very fast force curve at every pixel in the image. The peak interaction force in each of these force curves is then used as the imaging feedback signal. Peak Force Tapping mode modulates the Z-piezo at ~2 kHz with a default Peak Force Amplitude of 150 nm (0-peak). All values above and below this 0-peak magnitude is recorded as a Peak Force Error from which image is constructed. Force-distance curves were recorded using a MFP3D-BIO instrument (Asylum Research Technology, Atomic Force F&E GmbH, Mannheim, Germany). Conical gold coated AFM-tips were purchased from Bruker (NPG, Bruker AXS, Palaiseau, France) with spring constants of about 120-150 pN/nm. Experiments were performed in PBS at different pH values (3, 5, 7, 9 and 10) and at room temperature. Adhesion forces, conformational characteristics of the adhesins and their dynamic behavior were measured by recording Force-Volume Images (FVI) consisting of a grid of 32-by-32 force curves obtained upon approach and subsequent retraction of the tip (Single Molecule Force Spectroscopy for SMFS experiments) at a loading rate in the range 10 4 -10 5 pN/s (variable retraction speed for DFS experiments).

In the SMFS experiments, the adhesins located on the gold-surfaces and on the biological samples were stretched upon removal of the chemically modified AFM tip away from the surface (see Figure S2). The obtained force versus distance curves were then analyzed on the basis of the WLC (Worm Like Chain) model. This model is the most suitable and frequently used to describe the extension of polypeptides. 74,75 The extension z of a macromolecule is related to the retraction force F adh via (eq.1):

( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - + - = - 4 1 1 4 2 adh c c p B L z L z l T k z F (eq.1)
where the persistence length l p is a direct measure of the chain stiffness, L c is the total contour length of the biomacromolecule and k B is the Boltzmann constant.

List of Figures

Figure 1 .Figure 2 .

 12 Figure 1. Morphological analyses and nanogold-mediated detection of Ag43 produced

Figure 3 .

 3 Figure 3. Adhesion forces and statistical analysis from WLC modeling.

Figure 4 .

 4 Figure 4. Dependence of the maximal molecular extension and rupture force on solution

Figure 5 .Figure 6 .Figure 7 .Figure 8 .

 5678 Figure 5. Effect of the solution pH on the catch-bond-like behavior of Ag43α.

Figure 9 .

 9 Figure 9. Schematic representation of the unfolding of cis-associated Ag43α

Table 1 .

 1 TABLES Strains and plasmids used in this work.

	Strain or plasmid	Relevant genotypic and phenotypic	Source or
		characteristics	reference
	Strains
	TG	TG1 cured of the F plasmid, proline auxotroph	77
	MG1655∆oxyR	MG1655 deleted for oxyR, constitutive
		production of Ag43, Km R	72
	MG1655kmPcLflu	flu (agn43) gene placed under the control of the
		Ag43 production, Km R kmPcLrbs cassette λP R promoter, constitutive	43
	TG_kmPcLflu-His	6x-His tag placed 6 amino-acids after the Ag43
		signal sequence cleavage site in strain TG with constitutive expression of flu (agn43), Km R	This study
	E2152	MG1655_gfp_∆fliE-
		R::cat_∆flu::Km_∆fimAICDFGH::zeo,
		GFP+, no flagella, no Ag43 protein, no type 1	32
		fimbriae, Amp R , Cm R , Km R , Zeo R
	E2498	MG1655_gfp_∆fliE-
		R::cat_∆fimAICDFGH::zeo_kmPcLflu,
		GFP+, no flagella, no type 1 fimbriae,	32
		constitutive Ag43, Amp R , Cm R , Km R , Zeo R
	Plasmids
	pKOBEG	pSC101 ts (replicates at 30°C), araC, arabinose-	78
		inducible λredγβα operon , Cm R

Table 2 .

 2 Values of the conformational parameters (number of rupture events per force curve, corresponds to the average value of N 1 and ** corresponds to the average value of N 2 . See text for further details.

	persistence length, contour length, rupture force and number of monomers) derived from
	WLC modeling (eq 1) at pH 7.4 for various retraction rates (indicated). Data were extracted

from a 5 µm × 5 µm Force Volume Image (FVI) consisting of 1024 force curves recorded at pH 7 over the range of pulling rate 100 -3200 nm/s. The double entries for persistence length, contour length and number of monomers originate from the bimodal distribution issued from the statistical analysis of Figure

3

and Figure

S4

.

*

Table 3 .

 3 Values of the number of monomers derived from WLC modelling (eq 1) for various pH values at a pulling rate of 400 nm/s. Data were extracted from a 5 µm × 5 µm Force Volume

	Image (FVI) consisting of 1024 force curves recorded over a pH range of 5-9 for a pulling
	rate of 400 nm/s.

The number of monomers in the polypeptide chains was then derived from the following equation (eq.2):

All the FVI were analyzed by mean of an automatic Matlab algorithm described elsewhere. 39

Electronic Supplementary Information (ESI)

Electronic Supplementary Information (ESI) available: This material is available free of charge via the Internet at http://www.rsc.org/.

Representation of the envisaged formation and unfolding of a dimer of Ag43α formed by cisinteractions. The bends of the twisted L-shaped proteins are in opposite direction a) or along the same direction b).


Joaquin Fontbona
email: fontbona@dim.uchile.cl.

Hélène Guérin
email: helene.guerin@univ-rennes1.fr.

Sylvie Méléard
email: sylvie.meleard@polytechnique.edu

Fondecyt Proyect

Measurability of optimal

Keywords: Landau type interacting particle systems, nonlinear white noise driven SDE, pathwise coupling, measurability of optimal transport, predictable transport process. MSC: 60K35, 49Q20, 82C40, 82C80, 60G07

published or not. The documents may come

Introduction and main statements

Consider the nonlinear diffusion processes in R d of the following type:

X t = X 0 + t 0 R d σ(X s -y)W P (dy, ds) + t 0 R d b(X s -y)P s (dy)ds (1)
where P t is the law of X t , and W P is a R d valued space-time white noise on [0, T ] × R d with independent coordinates, each of which having covariance measure P t (dy) ⊗ dt.

The nonlinear process [START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF] was introduced by Funaki [START_REF] Funaki | A certain class of diffusion processes associated with non linear parabolic equations[END_REF], who obtained existence and uniqueness results for Lipschitz coefficients σ : R d → R d⊗d and b : R d → R d , see also Guerin [START_REF] Guérin | Solving Landau equation for some soft potentials through a probabilistic approach[END_REF] for a different approach. It has an important interpretation in terms of partial differential equations issued from kinetic theory. More precisely, for a specific choice of coefficients σ and b, the laws (P t ) t are a weak solution of the spatially homogeneous Landau (also called Fokker-Planck-Landau) equations for Maxwell potential:

∂f ∂t (t, v) = 1 2 d i,j=1 ∂ ∂v i R d a ij (v -v * ) f (t, v * ) ∂f ∂v j (t, v) -f (t, v) ∂f ∂v * j (t, v * ) dv * , (2)
with a ij (v) := (σσ * ) ij (v) = |v| 2 δ ij -v i v j and b i (v) = ∇ • a i• (v).
The equations (2) model collisions of particles in a plasma and can be obtained as limit of the Boltzmann equations when collisions become grazing, see Funaki [START_REF] Funaki | The diffusion approximation of the spatially homogeneous Boltzmann equation[END_REF], Goudon [START_REF] Goudon | Sur l'équation de Boltzmann homogène et sa relation avec l'équation de Landau-Fokker-Planck: influence des collisions rasantes[END_REF], Villani [START_REF] Villani | On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations[END_REF] [START_REF] Villani | On the spatially homogeneous Landau equation for Maxwellian molecules[END_REF] and Guérin-Méléard [START_REF] Guérin | Convergence from Boltzmann to Landau processes with soft potential and particle approximations[END_REF].

In this work, we shall prove the convergence in law of an easily simulable mean field interacting particle system towards the nonlinear process (1) at an explicit pathwise rate. This problem is of great interest in order to construct a tractable simulation algorithm for the law P t and thus, in particular, for solutions f of equation [START_REF] Cordier | A brief review on numerical methods for the collisions operators[END_REF]. To our knowledge, there is no result on convergence rates of the deterministic numerical methods used at present for the Landau equation, which are reviewed in [START_REF] Cordier | A brief review on numerical methods for the collisions operators[END_REF]. The interest of our approach is that it is based on the diffusive nature of the equation, and that it addresses a large class of nonlinear processes. The fact that we want to deal with simulable systems will necessitate a coupling between finite dimensional and infinite dimensional stochastic processes. We shall introduce a coupling argument based on new results on measurability of the optimal mass transportation problem.

We consider a particle system which is naturally related to the nonlinear process. Indeed, notice that the diffusion matrix associated with [START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF] is defined on R d by a(x, P t ) := R d σ(x -y)σ * (x -y)P t (dy) = [(σσ * ) * P t ](x).

(3) Thus, if in order to approximate the white noise driven stochastic differential equation (1), we heuristically replace P t in (3) by an empirical measure of n ∈ N * particles in R d , we are led to consider the following system driven by n 2 independent Brownian motions (B ik ):

X i,n t = X i 0 + 1 √ n t 0 n k=1 σ(X i,n s -X k,n s )dB ik s + 1 n t 0 n k=1 b(X i,n s -X k,n s )ds, i = 1, . . . , n. (4)
To be more precise, if

µ n t = 1 n n i=1 δ X i,n t
is the empirical measure of the system, the mappings

f (t, ω, x) → 1 √ n t 0 n k=1 f (s, ω, X k,n s )dB ik s , i = 1, . . . , n, (5)
define (for suitably measurable functions f ) orthogonal martingale measures in the sense of Walsh [START_REF] Walsh | An introduction to stochastic partial differential equations, École d'été de Probabilités de Saint-Flour XIV[END_REF], with covariance measure µ n t ⊗ dt.

By adapting techniques of Méléard-Roelly [START_REF] Méléard | Système de particules et mesuresmartingales: un théorème de propagation du chaos[END_REF] based on martingale problems, one can show propagation of chaos for system (4) with as limit the process (1). This says in particular that the covariance measure of (5) converges in law to P t ⊗ dt when n goes to infinity. But in turn, the arguments of [START_REF] Méléard | Système de particules et mesuresmartingales: un théorème de propagation du chaos[END_REF] do not give any information about speed of convergence.

To estimate the distance between the law of the particles and the law of the nonlinear process, we need to construct a significant coupling between finitely many Brownian motions and the white noises processes. This problem is much more subtle than in the McKean-Vlasov model (cf. Sznitman [START_REF] Sznitman | Topics in propagation of chaos[END_REF] or Méléard [START_REF] Méléard | Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models[END_REF]), where each particle is coupled with a limiting process through a single Brownian motion that drives them both. The well known

1 √
n -convergence rate in that model is consequence of the standard L 2 -law of large numbers in R d and of the fact that the diffusion and drift coefficients of the nonlinear process depend linearly on the limiting law through expectations with respect to it. In the present Landau model, we have to deal with the space-time random fields [START_REF] Goudon | Sur l'équation de Boltzmann homogène et sa relation avec l'équation de Landau-Fokker-Planck: influence des collisions rasantes[END_REF], which have fluctuations of constant order in n. This is also reflected in the fact that it is the squared diffusion matrix of (1), that depends linearly on P t (see ( 3)). It is hence not clear where a convergence rate can be deduced from.

Let X i ,i = 1, . . . , n be n independent copies of the nonlinear process in some probability space, and ν n t their empirical measure at time t (observe that it samples P t ). We shall construct particles (4) on the same probability space, in such way that they will converge pathwise in L 2 on finite time intervals, at the same rate at which the Wasserstein distance W 2 between P t and ν n t goes to 0. Let us state our main result on the process (1):

Theorem 1.1. Let n ∈ N and assume usual Lipschitz hypothesis on σ and b, and that the law P 0 of X i 0 has finite second order moment. Assume moreover that P t has a density with respect to Lebesgue measure for each t > 0. Then, in the same probability space as (X 1 , . . . , X n ) there exist independent standard Brownian motions (B ik ) 1≤i,k≤n such that the particle system (X i,n ) n i=1 defined in (4) satisfies

E sup t∈[0,T ] |X i,n t -X i t | 2 ≤ C exp(C T ) T 0 E(W 2 2 (ν n s , P s ))ds
for constants C, C that do not depend on n.

Thanks to available convergence results for empirical measures of i.i.d samples (see e.g. [START_REF] Rachev | Mass Transportation Problems, Volumes I and II[END_REF]), Theorem 1.1 will allow us to obtain, under some additional moment assumptions on P 0 , the speed of convergence n -2

d+4 for the pathwise law of the system (see Corollary 6.2). We remark that the absolute continuity condition of Theorem 1.1 can be obtained under non-degeneracy of the matrix σσ * by using for instance Malliavin calculus [START_REF] Nualart | The Malliavin calculus and related topics[END_REF]; it is also true for the specific coefficients of the Landau equation (2) despite their degeneracy, and for some generalizations (see Guérin [START_REF] Guérin | Existence and regularity of a weak function-solution for some Landau equation with a stochastic approach[END_REF]).

The proof of Theorem 1.1 relies on new results on the optimal mass transportation problem. For general background on the theory of mass transportation, we refer to Villani [START_REF] Villani | Topics in Optimal Transportation[END_REF]. Recall that if µ and ν are probability measures in R d with finite second moment, the first of them having a density, then the optimal mass transportation problem with quadratic cost between µ and ν has a unique solution, which is a probability measure on R 2d of the form π(dx, dy) = µ(dx)δ T (x) (dy) . The so-called Brenier or optimal transport map T (x) is (µ a.s. equal to) the gradient of some convex function in R d , and pushes forward µ to ν.

Let now W i P be the white noise process driving the i-th nonlinear process X i . The key idea in Theorem 1.1 will be to construct Brownian motions (B ik ) k=1...n in an "optimal" pathwise way from W i P . Heuristically, this will consist in pushing forward the martingale measure W i P through the Brenier maps T t,ω,n (x) realizing the optimal transport between P t and ν n t (ω) (this is the reason for the absolute continuity assumption on P t ). But to give such a construction a rigorous sense, we must make sure that we can compute stochastic integrals of T t,ω,n (x) with respect to W i P (dx, dt). From the basic definition of stochastic integration with respect to space-time white noise (cf. [START_REF] Walsh | An introduction to stochastic partial differential equations, École d'été de Probabilités de Saint-Flour XIV[END_REF]), this requires the existence of a measurable version of (t, ω, x) → T t,ω,n (x) being moreover predictable in (t, ω). A striking problem then is that no available result in the mass transportation theory can provide any information about joint measurability properties of the optimal transport map, with respect to the space variable and some parameter making the marginals vary. Nevertheless, we will show that a suitable "predictable transportation process" exists: Theorem 1.2. There exists a measurable process (t, ω, x) → T n (t, ω, x) that is predictable in (t, ω) with respect to the filtration associated to (W 1 P , . . . , W n P ) and (X 1 0 , . . . , X n 0 ), and such that for dt ⊗ P(dω) almost every (t, ω),

T n (t, ω, x) = T t,ω,n (x) P t (dx)-almost surely.
This statement is consequence of a general abstract result about "measurability" of the mass transportation problem. To be more explicit, recall that the optimality of a transfert plan π is determined by its support (it is equivalent to the support being cyclically monotone, see McCann [START_REF] Mccann | Existence and uniqueness of monotone measure-preserving maps[END_REF] or Villani [START_REF] Villani | Topics in Optimal Transportation[END_REF]). On the other hand, without assumptions (besides moments) on the marginals µ and ν, the solution π of the mass transportation problem may not be unique. A basic question then is how to formulate, in a general setting, the adequate property of "measurability" of the solution(s) π with respect to the data (µ, ν). As we shall see, the natural formulation requires to introduce notions and techniques from set-valued analysis. Then, we shall prove the following Theorem 1.3. Let P 2 (R d ) be the space of Borel probability measures in R 2 with finite second order moment, endowed with the Wasserstein distance and its Borel σ-field. Denote by Π * (µ, ν) the set of solutions of the mass transportation problem with quadratic cost associated with (µ, ν) ∈ (P 2 (R d )) 2 . The function assigning to (µ, ν) the set of R 2d :

π∈Π * (µ,ν) supp(π),
is measurable in the sense of set-valued mappings.

In particular, this ensures that if µ λ and ν λ vary in a measurable way with respect to some parameter λ, so that in each of the associated optimal transportation problems uniqueness holds, then the support of the solution π λ also "varies" in a measurable way. This will be the key to our results.

The rest of this work is organized as follows. In Section 2 we review the Wasserstein distance and the mass transportation problem with quadratic cost in R d (in particular the characterization of its minimizers). In Section 3 we prove Theorem 1.3 and a consequence needed to prove Theorem 1.1. In Section 4, we state some properties about process (1) and we heuristically describe our coupling between space-time white noises and Brownian motions. In Section 5 we construct the "predictable transportation process" of Theorem 1.2 needed to rigorously define the coupling. Section 6 is devoted to complete the proof of Theorem 1.1 and to obtain explicit convergence rates.

The mass transportation problem with quadratic cost in R d and the Wasserstein distance

We denote the space of Borel probability measures in R d by P(R d ), and by P 2 (R d ) the subspace of probability measures having finite second order moment. Given π ∈ P 2 (R 2d ), we respectively denote by π 1 and π 2 its first and second marginals on R d . On the other hand, for any two probability measures µ, ν ∈ P 2 (R d ) and π ∈ P 2 (R 2d ), we write π < µ ν if π 1 = µ and π 2 = ν. Such π is refereed to as a "transfert plan" between µ and ν.

Definition 2.1. The Wassertein distance W 2 on P 2 (R d ) is defined by

W 2 2 (µ, ν) := inf π< µ ν R 2 |x -y| 2 π(dx, dy).
Then, (P 2 (R d ), W 2 ) is a Polish space, see e.g. Rachev and Rüschendorf [START_REF] Rachev | Mass Transportation Problems, Volumes I and II[END_REF]. The topology is stronger that the usual weak topology. More precisely, one has the following result (see for instance Villani, [START_REF] Villani | Topics in Optimal Transportation[END_REF] Theorem 7.12)

Theorem 2.2. Let µ n , µ ∈ P(R d ).
The following are then equivalent:

i) W 2 (µ n , µ) → 0 when n → ∞.
ii) µ n converges weakly to µ and

R d |x| 2 µ n (dx) → R d |x| 2 µ(dx).
iii) We have

R d ϕ(x)µ n (dx) → R d ϕ(x)µ(dx)
for all continuous function ϕ :

R d → R such that |ϕ(x)| ≤ C(1+|x| 2 ) for some C ∈ R.
We shall denote by L the mapping L : P 2 (R 2d ) → R defined by

L(π) = R 2
|x -y| 2 π(dx, dy).

Remark 2.3. It is not hard to check that L is lower semi continuous (l.s.c) for the weak topology. Moreover, L is continuous for the Wasserstein topology in P 2 (R 2d ) by part iii) of Theorem 2.2.

Fix now µ, ν ∈ P 2 (R d ), and denote by Π * (µ, ν) the subset of P 2 (R 2d ) of minimizers of the Monge-Kantorovich transportation problem with quadratic cost for the pair of marginals (µ, ν) . That is, Π * (µ, ν) := argmin π< µ ν L(π). It is well known that Π * (µ, ν) is non-empty. Indeed, it is not hard to see that for the weak topology, {π ∈ P 2 (R 2d ) : π < µ ν } is a compact set, and the lower semi-continuity of L implies the existence of minimizers (see e.g. [START_REF] Villani | Topics in Optimal Transportation[END_REF] Chapter 1 for details).

We shall next recall the characterization of minimizers of the transportation problem with quadratic cost. We need the notion of sub-differential of a convex function:

Definition 2.4. Let ϕ : A ⊂ R d →] -∞, ∞] be a proper (i.e. ϕ ≡ +∞) lower semi- continuous (l.s.c) convex function. The sub-differential of ϕ at x is ∂ϕ(x) = {y ∈ R d : ϕ(z) ≥ ϕ(x) + y, z -x , ∀z ∈ R d }.
Elements of ∂ϕ(x) are called sub-gradients of ϕ at point x. The graph of ∂ϕ is

Gr(∂ϕ) = {(x, y) ∈ R 2d : y ∈ ∂ϕ(x)}
and it is a closed set.

Recall that ϕ is differentiable at x if and only if ∂ϕ(x) is a singleton (in which case ∂ϕ(x) = {∇ϕ(x)}). Also, the set {x ∈ R d : ϕ is differentiable at x} is borelian, see e.g. McCann [START_REF] Mccann | Existence and uniqueness of monotone measure-preserving maps[END_REF].

We next summarize results in pioneer works in this domain, Knott-Smith [START_REF] Knott | On the optimal mapping of distributions[END_REF], Brenier [START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF] and McCann [START_REF] Mccann | Existence and uniqueness of monotone measure-preserving maps[END_REF], Rachev and Rüschendorf [START_REF] Rachev | Mass Transportation Problems, Volumes I and II[END_REF]. See also Villani [START_REF] Villani | Topics in Optimal Transportation[END_REF] for a complete discussion on these questions, proofs and background.

Theorem 2.5. Let µ, ν ∈ P(R d ) and π < µ ν be a transfert plan. We have a) π ∈ Π * (µ, ν) if and only if there exists a proper l.s.c. convex function ϕ such that

supp(π) ⊂ Gr(∂ϕ) or, equivalently π({(x, y) ∈ R 2 : y ∈ ∂ϕ(x)}) = 1.
b) Assume that µ does not charge sets of Hausdorff dimension less or equal than d -1 and that π ∈ Π * (µ, ν). Then, i) the set {x ∈ R d : ϕ is not differentiable at x} has null µ-measure.

ii) We have π(dx, dy) = µ(dx) ⊗ δ ∇ϕ(x) (dy).

ii) If T is a measurable mapping such that π(dx, dy) = µ(dx) ⊗ δ T (x) (dy), then

T (x) = ∇ϕ(x) , µ(dx) -a.s.. iii) π ∈ Π * (µ, ν) is unique.
This result will be useful later in the particular case when the measure µ is absolutely continuous with respect to Lebesgue measure.

Measurability of the mass transportation problem

We now introduce the basic notions on "multi-applications" or "set-valued mappings" that we need to prove Theorem 1.3. For general background, we refer the reader to Appendix A in Rockafellar and Wets [START_REF] Rockafeller | Variational Analysis[END_REF].

Definition 3.1. Let X, Y be two sets.

i) A function S on X taking values in the set of subsets of Y is called a set-valued mapping or multi-application. We write S : X ⇒ Y .

ii) For any A ⊂ Y , the inverse image of A through S is the set

S -1 (A) := {x ∈ X : S(x) ∩ A = ∅}.
iii) If (X, A) is a measurable space and (Y, Θ) a topological space, we say that S : X ⇒ Y is measurable if for all θ ∈ Θ, S -1 (θ) ∈ A.

(Of course, if S(x) = {s(x)} is singleton for all x, measurability of S is equivalent to that of s. )

Consider P 2 (R d ) endowed with the Wasserstein distance and the Borel σ-field. We define a set-valued mapping Ψ :

(P 2 (R d )) 2 ⇒ R 2d by Ψ(µ, ν) := {(x, y) : ∃π ∈ Π * (µ, ν) s.t. (x, y) ∈ supp(π)}.
Our goal is to prove that Ψ is measurable. We shall need some further notions on set-valued mappings.

Definition 3.2. Let X be a set, and (Y, Ξ) and (Z, Θ) be topological spaces.

i) A set-valued mapping S :

X ⇒ Y is closed-valued if for all x ∈ X, S(x) is a closed set of (Y, Ξ). ii) A set-valued mapping U : Y ⇒ Z is inner semicontinous (i.s.c) if for all θ ∈ Θ, S -1 (θ) ∈ Ξ
The following results can be found in Appendix A of [START_REF] Rockafeller | Variational Analysis[END_REF], in the case of set-valued mappings in R d . For completeness we provide proofs in a more general context.

Lemma 3.3. Let (X, A) be a measurable space and (Y, Ξ) a topological space.

i) S : X ⇒ Y is measurable if and only if the closed-valued mapping x ⇒ Cl(S(x)) is measurable, where Cl(S(x)) is the topological closure of the set S(x).

ii) Assume that (Y, d) is a separable metric space and that S :

X ⇒ Y is closed-valued.
Then, S is measurable if and only if for all closed set F of Y ,

S -1 (F ) ∈ A.
iii) Let (Y, Ξ) and (Z, Θ) be topological spaces, S : X ⇒ Y be measurable and U : Y ⇒ Z be i.s.c. Then, the multi-application U • S : X ⇒ Z, defined by

U • S(x) := y∈S(x) U (y) is measurable. Proof i) For any open set θ ∈ Ξ, S(x) ∩ θ = ∅ if and only if Cl(S(x)) ∩ θ = ∅.
ii) "Only if" part: since Y is a metric space, we use that every closed set F is the intersection of some countable collection of open sets (θ n ). Therefore,

{x ∈ X : S(x) ∩ F = ∅} = n∈N {x ∈ X : S(x) ∩ θ n = ∅} ∈ A.
"If" part: (Y, d) being separable, we can express every open set θ as the union of some countable collection (B n ) of closed balls. We then have that

{x ∈ X : S(x) ∩ θ = ∅} = n∈N {x ∈ X : S(x) ∩ B n = ∅} ∈ A.
iii) Straightforward:

(U • S) -1 (θ) = {x ∈ X : ∪ y∈S(x) U (y) ∩ θ = ∅} = {x ∈ X : ∃y ∈ S(x) s.t. U (y) ∩ θ = ∅} = {x ∈ X : S(x) ∩ (U -1 (θ)) = ∅}.
The function U being i.s.c., U -1 (θ) belongs to Ξ, which allows us to conclude.

Now we can proceed to the

Proof of Theorem 1.3

We observe first that Ψ(µ, ν) = U • S(µ, ν), where S and U are the set valued mappings respectively defined by (µ, ν) ⇒ S(µ, ν) := Π * (µ, ν)

and

U : P 2 (R 2d ) ⇒ R d by U (π) := supp(π)
We will therefore split the proof in several parts: a) S is a closed valued mapping First notice that π → π i is continuous for the Wasserstein topology. Indeed, W 2 (π n , π) → 0 implies that π n converges weakly to π, and then π n i converges weakly to π i for i = 1, 2. Moreover, we have

R d |x| 2 π n 1 (dx) = R 2d |x| 2 π n (dx, dy) → R 2d |x| 2 π(dx, dy) = R d |x| 2 π 1 (
dx) by Theorem 2.2, and then the asserted continuity follows. Consequently, π → W 2 (π 1 , π 2 ) too is continuous. Therefore,

Π * (µ, ν) = {π : π < µ ν } ∩ {π : L(π) -W 2 (π 1 , π 2 ) = 0}
is the intersection of two closed sets P 2 (R 2d ).

b) Inverse images through S of closed sets are closed sets Let F ⊂ P 2 (R d ) be a closed set and (µ n , ν n ) ∈ S -1 (F ), n ∈ N , be a sequence converging to (µ, ν) in (P 2 (R d )) 2 . Then, µ n → µ and ν n → ν weakly, and (µ n ) and (ν n ) are tight. But since (µ n , ν n ) ∈ S -1 (F ) for each n, there exists π n s.t. π n < µ n ν n , and then (π n ) too is tight (by considering products of compact sets). Let (π n k ) be a weakly convergent subsequence with limit π. Then, clearly π < µ ν . We will prove that L(π) = W 2 (µ, ν) and that π ∈ F , which will mean that (µ, ν) ∈ S -1 (F ) and finish the proof.

We have

R 2d |x| 2 + |y| 2 π n k (dx, dy) = R d |x| 2 µ n k (dx) + R d |y| 2 ν n k (dy) → R d |x| 2 µ(dx) + R d |y| 2 ν(dy) = R 2d |x| 2 + |y| 2 π(dx, dy),
which implies that W 2 (π n , π) → 0 and π ∈ F . Finally, by the continuity of π → L(π) -W 2 (π 1 , π 2 ) we get that

0 = L(π n k ) -W 2 (π n k 1 , π n k 2 ) = L(π) -W 2 (µ, ν).
c) The mapping U is i.s.c. Let θ be an open set of R 2d . We must check that

{π ∈ P 2 (R 2d ) : supp(π) ∩ θ = ∅} = {π ∈ P 2 (R 2d ) : π(θ) > 0}
is open, or equivalently, that

{π ∈ P 2 (R 2d ) : π(θ) = 0} is closed in P 2 (R 2d
). Assume that π, π n ∈ P 2 (R 2d ), with π n such that π n (θ) = 0 for all n ∈ N, and moreover that W 2 (π n , π) → 0. Then π n converges weakly to π, and so by the Portemanteau theorem, we have

0 = lim inf n π n (θ) ≥ π(θ).

d) Conclusion

By parts a) and b) and Lemma 3.3 ii) we get that S is measurable. By c) and Lemma 3.3 iii) U • S is measurable and the proof is finished.

The following corollary will be useful in the specific setting needed to prove Theorem 1.1:

Corollary 3.4. Let (E, Σ) be a measurable space, and λ ∈ E → (µ λ , ν λ ) ∈ (P 2 (R d )) 2 and ξ : E → R d be measurable functions. Then, the set

{(λ, x) : (x, ξ(λ)) ∈ Cl(Ψ)(µ λ , ν λ )} belongs to Σ ⊗ B(R d )
Proof By Lemma 3.3 i) and Theorem 1.3 we get that Cl(Ψ) is measurable. Moreover, it is not hard to check that the mapping

(λ, x) ⇒ Cl(Ψ)(µ λ , ν λ ) -(x, ξ(λ))
is measurable and closed-valued. Then, we just have to notice that

(x, ξ(λ)) ∈ Cl(Ψ)(µ λ , ν λ ) if and only if [Cl(Ψ)(µ λ , ν λ ) -(x, ξ(λ))] ∩ C = ∅
for the closed set C = {0}.

A coupling between space-time white noise and Brownian motions via optimal transport

In all the sequel, we refer the reader to Walsh [START_REF] Walsh | An introduction to stochastic partial differential equations, École d'été de Probabilités de Saint-Flour XIV[END_REF] for background on space-time white noise processes and stochastic integration with respect to martingale measures. Assume that σ : R d → R d⊗d and b : R d → R d are Lipschitz continuous and with linear growth. Then, by results of [START_REF] Funaki | A certain class of diffusion processes associated with non linear parabolic equations[END_REF] or [START_REF] Guérin | Solving Landau equation for some soft potentials through a probabilistic approach[END_REF] we can construct in some probability space (Ω, F, P) a sequence (X i ) i∈N of independent copies of the nonlinear processes,

X i t = X i 0 + t 0 R d σ(X i s -y)W i P (dy, ds) + t 0 R d b(X i s -y)P s (dy)ds, ( 6
)
where the W i P are independent space-time R d -valued white noises defined on [0, ∞) × R d . Each of the d (independent) coordinates of W i P has covariance measure P t (dy) ⊗ dt, where P t is the law of X t . The initial conditions (X 1 0 , . . . , X n 0 , . . . ) are independent and identically distributed with law P 0 , and independent of the white noises. The pathwise law of X i is denoted by P , and it is uniquely determined.

Denote by F n t the complete right continuous σ-field generated by

{(W 1 P ([0, s] × A 1 ), . . . , W n P ([0, s] × A n )) : 0 ≤ s ≤ t, A i ∈ B(R d )}
and (X 1 0 , . . . , X n 0 ). We also denote by Pred n the predictable field generated by continuous (F n t )-adapted processes. In what follows, we fix a finite time horizon T > 0. Under usual Lipschitz assumptions on the coefficients, there is propagation of the moments of the law P 0 , as proved in Guérin [START_REF] Guérin | Solving Landau equation for some soft potentials through a probabilistic approach[END_REF].

Lemma 4.1. If E(|X 0 | k ) < ∞ for some k ≥ 2, then E sup t∈[0,T ] |X t | k < ∞.
The continuity of X and the previous uniform bound imply that t → R d |x| k P t (dx) is continuous.

Throughout the sequel, the assumptions of Theorem 1.1 on P 0 and P t are enforced, in particular, the condition E(sup t∈[0,T ] |X t | 2 ) < ∞ will hold by the previous lemma.

We shall now present the main idea of the coupling we introduce to prove Theorem 1.1. Basically, this consists in constructing for each n, n 2 Brownian motions in a pathwise way, from the realizations of the n white noises (W 1 P , . . . , W n P ). The key for that will be to use the optimal transport maps between the marginal P t of the nonlinear process and the empirical measures of samples of that law. More precisely, write

ν n t := 1 n n i=1 δ X i t
and notice that for each ω ∈ Ω, (ν n t , 0 ≤ t ≤ T ) is an element of C([0, T ], P 2 (R d )). Thus, for each t ∈ [0, T ], n ∈ N and ω, and we can consider the optimal coupling problem with quadratic cost between ν n t (ω) and P t ,

inf π< P t ν n t (ω) R d ×R d
|x -y| 2 π(dx, dy) .

By the assumption on P t and Theorem 2.5, the following properties hold for each fixed pair (t, ω) ∈]0, T ] × Ω:

Lemma 4.2.
a) There exists a unique π t,ω,n , such that

W 2 2 (P t , ν n t (ω)) = R d ×R d |x -y| 2 π t,ω,n (

dxdy).

b) There is a P t (dx) -a.e. unique measurable function T t,ω,n : R d → R d such that π t,ω,n (dx, dy) = δ T t,ω,n (x) (dy)P t (dx).

In particular, under P t (dx) the law of T t,ω,n (x) is ν n t (ω).

c) We have

W 2 2 (P t , ν n t (ω)) = R 2 |x -T t,ω,n (x)| 2 P t (dx).
We would like to construct n 2 independent Brownian motions by "transporting" the n independent white noises (W 1 P , . . . , W n P ) through the transport mappings T s,ω,n (x). As pointed out in the introduction, to do so we must at least be able to define stochastic integrals of functions of the form (t, ω, x) → f (T t,ω,n (x)), with respect to the white noise processes. The existence of a version T n (t, ω, x) of T t,ω,n (x) having good enough properties, will be established in next section, when we shall prove Theorem 1.2. Before doing so, we observe that if Theorem 1.2 holds, then the following processes B ik t = B ik,n t will be well defined from [START_REF] Guérin | Existence and regularity of a weak function-solution for some Landau equation with a stochastic approach[END_REF].

Proposition 4.3. For each n ∈ N * , define B ik,n t (ω) := √ n t 0 R d 1 {T n (s,ω,x)=X k s (ω)} W i P (dx, ds), i, k = 1 . . . n (7)
Then, (B ik,n ) 1≤i,k≤n are n 2 independent standard Brownian motions in R d .

These are the right Brownian motions we need to construct (4). The proof of Proposition 4.3 will given in Section 6.

Construction of the predictable "transport process"

Our goal now in this section is to show that for each n ∈ N * , there exists a process (t, ω, x) → T n (t, ω, x) defined P(dω) ⊗ dt ⊗ P t (dx)-almost everywhere, which is measurable with respect to Pred n ⊗ B(R d ), and such that:

for dt ⊗ P(dω) almost every (t, ω),

T n (t, ω, x) = T t,ω,n (x) P t (dx)-almost surely .

Since (X 1 , . . . , X n ) are independent copies of the nonlinear process and P t = law(X i t ) has a density with respect to Lebesgue measure, for each t ∈ [0, T ] we have that

P(∃i = j : X i t = X j t ) = 0.
Notice also that for fixed (i, j) with i = j the following set

{(t, ω) : X i t (ω) = X j t (ω)} belongs to Pred n since (t, ω) → |X i t (ω) -X j t (ω)
| is adapted and continuous in t. By Fubini's theorem we then see that

[0,T ]×Ω 1 {X i t =X j t } (t, ω)P(dω) ⊗ dt = 0 Remark 5.1. Consequently, there is a predictable set of [0, T ] × Ω, Ω T ∈ Pred n
of full P(dω) ⊗ dt-measure and such that for all (t, ω) ∈ Ω T , X i t (ω) = X j t (ω) for all i, j ∈ {1, . . . , n}.

Let us denote by (Pred n ) the σ-field Pred n restricted to Ω T .

Recall that for each (t, ω), the set of solutions Π * P t , 1 n n i=1 δ X i t (ω) of the optimal transport problem between P t and 1 n n i=1 δ X i t (ω) is a singleton that we have denoted by π t,ω,n . Let us define now the sets

A i,n := (t, ω, x) ∈ Ω T × R d : (x, X i t (ω)) ∈ supp(π t,ω,n ) , i = 1, . . . , n.
The sets A i,n are predictable, as proved in the following lemma.

Lemma 5.2. We have

A i,n ∈ (Pred n ) ⊗ B(R d ).
Proof Observe that the deterministic process (t, ω)

→ P t ∈ P 2 (R d ) is Pred n -measurable. Indeed, if (f n ) n∈N\{0,1}
is a countable dense subset of the space of continuous functions in R d with compact support, and f 0 (x) = 1,f 1 (x) = |x| 2 , then the topology of P 2 (R d ) is generated by the real mappings m → f n (x)m(dx). It is therefore enough that (t, ω) → f n (x)P t (dx) be Pred n -measurable, which is clear since t → P t is continuous. Next we will apply Corollary 3.4 to the measurable space (E, Σ) = (Ω T , (Pred n ) ), λ = (t, ω), and the (Pred n ) -measurable functions given by

(t, ω) →   P t , 1 n n j=1 δ X j t (ω)   ∈ P 2 (R 2d ) and (t, ω) → ξ i (t, ω) = X i t (ω) ∈ R d .
For each (t, ω) ∈ Ω T , with Ψ denoting the multi-application defined in Theorem 1.3, we simply have in the current setting that

Ψ   P t , 1 n n j=1 δ X j t (ω)   = supp(π t,ω,n ).
Corollary 3.4 implies the result.

Recall from basic measure theory that if E 1 and E 2 are measurable spaces and A ⊆ E 1 × E 2 is an element of their product σ-field, then, for each

λ 1 ∈ E 1 , the fiber of A at λ 1 is the set [A] λ 1 := {λ ∈ E 2 : (λ 1 , λ) ∈ A},
and it is always measurable in E 2 .

We can now proceed to the Proof of Theorem 1.2:

We split the proof in several parts.

a) The sets A i,n , i = 1 . . . n form a partition of Ω T × R d up to P(dω) ⊗ dt ⊗ P t (dx)-null sets.

For i = j write

A ij,n := {(t, ω, x) ∈ (Ω T × R d ) : (x, X i t (ω)) ∈ supp(π t,ω,n ) and (x, X j t (ω)) ∈ supp(π t,ω,n )} = A i,n ∩ A j,n , and denote by [A ij,n ] (t,ω) := {x ∈ R d : (t, ω, x) ∈ A ij,n } ∈ B(R d ) the fiber of A ij,n at (t, ω) ∈ Ω T . Then, we have P t ([A ij,n ] (t,ω) ) = P t ({x ∈ R d : (x, X i t (ω)), (x, X j t (ω)) ∈ supp(π t,ω,n )}) ≤ P t ({x ∈ R d : X i t (ω), X j t (ω) ∈ ∂ϕ t,ω,n (x)}),
where ϕ t,ω,n is a proper l.s.c. convex function given by Theorem 2.5 a). But since (t, ω) ∈ Ω T , we have X i t (ω) = X j t (ω), and so

X i t (ω), X j t (ω) ∈ ∂ϕ t,ω,n (x) =⇒ ϕ is not differentiable in x.
We obtain by Theorem 2.5 b) that P t ([A ij,n ] (t,ω) ) = 0, and then

E [0,T ]×Ω×R d 1 A i,n ∩A j,n (t, ω, x)P t (dx)dt = 0.
On the other hand, since T t,ω,n (x) ∈ {X 1 t (ω), . . . , X n t (ω)} P t (dx) a.s., we have for all (t, ω) that

P t   n i=1 A i,n c (t,ω)   = P t ({x ∈ R d : for all i = 1, . . . , n, (x, X i t (ω)) ∈ supp(π t,ω,n )}) ≤ P t ({x ∈ R d : (x, T t,ω,n (x)) ∈ supp(π t,ω,n )}) = π t,ω,n (supp(π t,ω,n ) c ) = 0
Defining the set

ΩT := (Ω T × R d )   n i=1 A i,n \   k =j A kj,n     ∈ Pred n ⊗ B(R d )
we deduce that P t [ Ωc T ] (t,ω) = 0 for all (t, ω) ∈ Ω T . Therefore,

E T 0 R d 1 Ωc T (t, ω, x)P t (dx)dt = Ω T P t [ Ωc T ] (t,ω) dt ⊗ P(dω) = 0,
and so ΩT has full P(dω) ⊗ dt ⊗ P t (dx)-measure. This proves assertion a).

We can now define a Pred n ⊗ B(R d )-measurable function by

T n (t, ω, x) := n i=1 1 A i,n ∩ ΩT (t, ω, x)X i t (ω). (8)
b) For P(dω) ⊗ dt almost every (t, ω), T n (t, ω, x) = T t,ω,n (x) holds P t (dx) almost surely. By Theorem 2.5, b), this is equivalent to prove that π t,ω,n (dx, dy) = P t (dx) ⊗ δ T n (t,ω,x) (dy) P(dω) ⊗ dt -a.e.

We fix now (t, ω) ∈ Ω T and C, D ∈ B(R d ).

We have by definition of T t,ω,n that

π t,ω,n (C × D) = R d 1 C (x)1 D (T t,ω,n (x))P t (dx) = R d 1 C∩[ ΩT ] (t,ω) (x)1 D (T t,ω,n (x))P t (dx),
the latter because P t [ Ωc ] (t,ω) = 0. Notice that on the other hand, by definition of A i,n , ΩT and T n , for all (t, ω, x) ∈ A i,n ∩ ΩT we have that {y : (x, y) ∈ supp(π t,ω,n )} = {X i t (ω)} = {T n (t, ω, x)}. This implies that ΩT ⊂ {(t, ω, x) ∈ Ω T × R d : {y : (x, y) ∈ supp(π t,ω,n )} is a singleton}. Now, let F t,ω ∈ B(R d ) be a measurable set with P t (F t,ω ) = 1 and such that T t,ω,n (x) = ∇ϕ t,ω,n (x) is defined for all x ∈ F t,ω . Then, on F t,ω ∩ [ ΩT ] (t,ω) it must hold that T n (t, ω, x) = T t,ω,n (x) = ∇ϕ t,ω,n (x), and we conclude that for all (t, ω) ∈ Ω T ,

π t,ω,n (C × D) = R d 1 C∩F t,ω ∩[ ΩT ] (t,ω) (x)1 D (T t,ω,n (x))P t (dx), = R d 1 C∩F t,ω ∩[ ΩT ] (t,ω) (x)1 D (T n (t, ω, x))P t (dx) = R d 1 C (x)1 D (T n (t, ω, x))P t (dx)
We point out that Theorem 1.2 implies Corollary 5.3. T n (t, ω, x) = T t,ω,n (x) holds P(dω) ⊗ dt ⊗ P t (dx)-almost surely . Consequently, T t,ω,n (x) is measurable with respect to the completed σ-field of Pred n ⊗B(R d ) with respect to P(dω) ⊗ dt ⊗ P t (dx).

6 Pathwise convergence and rates for stochastic particle systems to Landau process for the m-th coordinate of the process B ik,n t in [START_REF] Guérin | Solving Landau equation for some soft potentials through a probabilistic approach[END_REF], which is a real valued continuous local martingale with respect to F n t (see [START_REF] Walsh | An introduction to stochastic partial differential equations, École d'été de Probabilités de Saint-Flour XIV[END_REF]). Then, we have that

B ik,n,m , B i k ,n,m t (ω) =nδ (i,m),(i ,m ) t 0 R d 1 A k,n ∩A k ,n ∩ ΩT (s, ω, x)P s (dx)ds = nδ (i,k,m),(i ,k ,m ) t 0 R d 1 A k,n ∩ ΩT (s, ω, x)P s (dx)ds,
by step (a) in the proof of Theorem 1.2. Now, for (s, ω) ∈ Ω T the points X 1 s (ω), . . . , X n s (ω) are all different, and consequently we have that

R d 1 A k,n ∩ ΩT (s, ω, x)P s (dx) =P s ({x : T n (s, ω, x) = X k s (ω)}) =P s ({x : T s,ω,n (x) = X k s (ω)}) =π s,ω,n ({(x, y) : y = X k s (ω)}) =ν n s (X k s (ω)) = 1 n
Thus, we have B ik,n,m , B i k ,n,m t = tδ (i,k,m),(i ,k ,m ) , and the result follows.

We now are ready to prove Theorem 1.1.

Proof of Theorem 1.1, a) Let us fix n ∈ N * , and define for i = 1, . . . n,

X i,n t = X i 0 + 1 √ n t 0 n k=1 σ(X i,n s -X k,n s )dB ik,n s + 1 n t 0 n k=1 b(X i,n s -X k,n s )ds
or equivalently, in an indistinguishable way,

X i,n t =X i 0 + t 0 R d n k=1 σ(X i,n s -X k,n s )1 A k,n (s, y)W i P (dy, ds) + t 0 R d n k=1 b(X i,n s -X k,n s )1 A k,n (s, y)P s (dy)ds
By standard arguments and the fact that the sets A k,n are disjoint (step (a) of the proof of Theorem 1.2), we have

E |X i,n t -X i t | 2 ≤ t 0 R d E n k=1 σ(X i,n s -X k,n s ) -σ(X i s -y) 2 1 A k,n (s, y) P s (dy)ds + t 0 R d E n k=1 b(X i,n s -X k,n s ) -b(X i s -y) 2 1 A k,n (s, y) P s (dy)ds (9)
The first term in the right hand side of ( 9) is bounded by

C t 0 R d E n k=1 σ(X i,n s -X k,n s ) -σ(X i s -X k,n s ) 2 1 A k,n (s, y) P s (dy)ds + C t 0 R d E n k=1
σ(X i s -X k,n s ) -σ(X i s -T n (s, y)) by exchangeability of ((X 1,n , X 1 ), . . . , (X n,n , X n )). A similar bound is obtained for the second term in [START_REF] Knott | On the optimal mapping of distributions[END_REF]. We deduce by Gronwall's lemma that .

E |X i,
The previous results are the first convergence rates obtained so far for stochastic particle systems of the "Landau type" (4), and they are not specific to the particular coefficients of the Landau equation [START_REF] Cordier | A brief review on numerical methods for the collisions operators[END_REF]. They justify the interest of the particle systems introduced in (4) and are the first step in the construction and the numerical study of a simulation algorithm for (P t ) t . We notice that since we deal with space-time random fields, the dependence of the results on the dimension d is somewhat expectable, as opposite to the situation in the McKean-Vlasov model. The techniques we have introduced provide some insight about that dependence.

  Proof of Proposition 4.3From the proof of Theorem 1.2, it is clear that integrals with respect to the measures 1 A k,n ∩ ΩT P t (dx) ⊗ dt and 1 A k,n P t (dx) ⊗ dt are indistinguishable. By considering quadratic variations, the same is seen to hold for the stochastic integrals with respect to 1 A k,n ∩ ΩT W i

		P (dx, dt)
	and 1 A k,n W i P (dx, dt). Write	B ik,n,m t

  E |T n (s, ω, y) -y| 2 P s (dy)ds

														2
														1 A k,n (s, y)	P s (dy)ds
	+ C	0	t	R d	E	n k=1	σ(X i s -T n (s, y)) -σ(X i s -y)	2 1 A k,n (s, y)	P s (dy)ds
	≤ C	t	E			n	X i,n s -X i s	2
	0	k=1 R t n
								E		X k,n
		0			R d		k=1
		0	t	R d	E	n k=1	|T n (s, ω, y) -y| 2 1 A k,n (s, y)	P s (dy)ds
	= C				t	E X i,n s -X i s	2 ds
					0
	+ C	0	t	E		n k=1	X k,n s	-X k s	2	R d	1 A k,n (s, y)P s (dy)	ds
			t
	+ C
		0			R d
					0	t	E X i,n s -X i s	2 ds + C	0	t	1 n	E	k=1 n	X k,n s	-X k s	2	ds
			t
	+ C
		0			R d
	= 2C				t	E X i,n s -X i s	2 ds + C	t	E W 2 2 (ν n s , P s ) ds
				0									0

d 1 A k,n (s, y)P s (dy) ds + C s -T n (s, y)) 2 1 A k,n (s,

y) P s (dy)ds + C E |T n (s, ω, y) -y| 2 P s (dy)ds = C

  By a little finer argument using a Burkholder-Davis-Gundy inequality, we can obtain as usual an estimate of the form|X i,n t -X i t | 2 ≤ C exp(C T )We recall a result proved in Rachev and Rüschendorf[START_REF] Rachev | Mass Transportation Problems, Volumes I and II[END_REF] giving L 2 -rates of convergence of empirical measures in the Wasserstein metric. Theorem 6.1. ([14] Theorem 10.2.1) Let µ a probability on R d and let Y 1 , Y 2 , . . . , Y n be independent identically distributed random variables with law µ. Let µ n be the empirical measure of these variables. Then, if µ has high enough finite absolute moments: c := R d |y| d+5 µ(dy) < ∞, there is a constant C depending only on c and on the dimension d, such that E W 2 2 (µ n , µ) ≤ Cn Denote by W 2 the Wasserstein distance between probability measures Q on the path spaceC T := C([0, T ], R d ), such that C T sup 0≤t≤T |x(t)| 2 Q(dx) < ∞.From the previous result and Lemma 4.1, it is simple to deduce the following Corollary 6.2. Let P be the pathwise law of the nonlinear process (1). Under the assumptions of Theorem 1.1 and moreover that R d |y| d+5 P 0 (dy) < ∞, we have thatW 2 2 (law(X 1,n ), P ) ≤ C T,d n

		T
	E sup	E W 2 2 (ν n s , P s ) ds
	t∈[0,T ]	0
		-2 d+4 .
		-2
		d+4

n t -X i t | 2 ≤ C exp(C T ) t 0 E(W 2 2 (ν n s , P s )) ds

Acknowledgements

The authors are very grateful to Roberto Cominetti for helpful suggestions about the theory of set-valued mappings.