Nicholas M Randell

Julia Rendon

Martine Demeunynck

Pierre-Alain Bayle

Serge Gambarelli

Vincent Artero

Jean-Marie Mouesca

Murielle Chavarot-Kerlidou
email: murielle.chavarot-kerlidou@cea.fr

Tuning the electron storage potential of a charge-photoaccumulating Ru II complex by a DFT-guided approach

Keywords: Charge photoaccumulation, computational chemistry, energy conversion, ruthenium chromophore

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In recent decades, as awareness of both anthropogenic climate change and shrinking petroleum supplies has grown, scientists have searched for ways to transition society away from its dependence on fossil fuels. This effort has accelerated the growth of the field of solar fuels research. [1][2][3][4] Solar fuels, such as H2, CO, and formic acid, can be produced via light-driven reduction processes using abundant feedstock chemicals such as CO2 and H2O. Most of these transformations involve the transfer of multiple electrons, and significant progress has been made in developing molecular catalysts able to perform such kinetically arduous multi-proton multielectron processes. [5][6][7][8] Widely-used photosensitizers (PS) for solar fuel production include Ru(II) [9][10][11] and Ir(III) [12,13] polypyridyl coordination complexes, porphyrins, [14][15][16] and organic push-pull dyes. [17][18][19] While these chromophores feature broad absorption bands with relatively high extinction coefficients in the visible region, they fundamentally generate one photoexcited electron at a time. This implies that, for catalysis to occur, the system needs to accumulate electrons generated by sequential light absorption steps; this leads to excessive charge build-up and generates intermediate species prone to quench the PS excited state by reverse electron transfer or by energy transfer. [20][21][22] In addition, the catalytic rate should be fast enough in order to avoid competing reactions, typically charge recombination. The latter however take place at very fast timescales (fs to ns) compared to catalysis (s to ms range for H2 production, [23] typically). This represents a major issue for solar fuel-producing systems, and developing novel systems specifically designed to optimizing this charge accumulation process can provide a way to address this problem.

In nature, photosynthesis decouples catalysis from light-driven processes thanks to specific cofactors, such as plastoquinones or NAD(P) + , stepping in to relay photogenerated electrons two by two. Notably, stable reduced derivatives (dihydroquinones and NADPH) are generated thanks to a storage mechanism involving proton-coupled electron transfers (PCET). Taking inspiration from these systems, more sophisticated light-harvesting units, capable of storing multiple electrons upon visible light irradiation, have been synthesized and studied. [20][21][22] Two conceptually different molecular designs have proven successful in storing multiple reducing equivalents in artificial photosynthetic systems: either the system assembles multiple chromophores each of which independently transfers electrons to the acceptor site (such as a bridging ligand between two chromophore units), [24][25][26][27][28][29][30][31][32][33][34][35][36][37] or it contains a single chromophore and relies on a sacrificial electron donor to perform multiple excitation-accumulation cycles. [38][39][40][41][42][43][44][45] In the latter case, either a single or multiple electron storage sites can be assembled adjacent to the chromophore. Typical lightharvesting units are porphyrins, [24,36] Ir(III) [37] or Ru(II) polypyridine [25][26][27][28][29][30][31][32][33][34][35][38][39][40][41][42][43][44][45] complexes, covalently linked to a wide variety of electron storage sites, such as a metal centre, [25] polyoxometalates, [13,36] naphthalene diimide, [35,45] perylene diimide, [24] anthraquinone, [30,33,34] or extended phenanthroline-based [26][27][28][29]44,46,47] ligands. Recently, Wenger and coworkers produced one of the first examples of the beneficial role played by electron photoaccumulation in photoredox catalysis. They used assemblies of Ru(II) chromophores with a central dibenzo [1,2]dithiin electron storage unit to accumulate two photogenerated electrons, further used to perform dithiolate/disulfide interconversion. [31,32] Among these examples, systems relying on a single PS unit to promote unidirectional charge separation and storage on a unique acceptor site are the most likely candidates for the integration of a charge photoaccumulation process into functional systems such as dye-sensitized photocathodes for solar fuels production. [19,48,49] In these devices, the light-harvesting moiety is anchored onto a p-type semiconductor film such as NiO, allowing fast reductive quenching of its excited state by hole injection into the semiconductor valence band upon visible light irradiation.

By virtue of the unidirectional charge transfer, the electron-storage site is distant from the semiconductor surface, thus preventing back-electron transfer. The electron storage unit could also be bound to a catalyst, to promote catalyst activation and solar fuels production. Molecular chargephotoaccumulating systems answering to this topological requirement are however rather scarce in the literature. [42][43][44][45] In 2018, we reported a mononuclear ruthenium tris-diimine complex with an extended phenazine-based ligand [1](PF6)2 (Scheme 1) which falls into the second category of chargeaccumulating PS. 50,51] The phenazine ligand features a tetracyclic pyridoquinolinone subunit inspired by the quinone-based electron storage and relay systems involved in photosynthesis. [52] We demonstrated that [1](PF6)2 is capable of storing two photogenerated electrons upon excitation of the complex with visible light in the presence of a sacrificial electron donor. In the absence of a source of protons, the first reduction of [1] 2+ occurs at -0.87 V and the second at -1.32 V vs. the ferrocenium/ferrocene (Fc + /Fc) redox couple; these potentials are low enough to donate electrons to many of the best-performing catalysts for H2 production. In the presence of proton donors, however, the reduction processes become coupled to proton transfers, yielding the stabilized doubly-reduced doubly-protonated species [1H2] 2+ . The occurrence of PCET processes strongly lowers the reducing power of the system, precluding the use of charge photoaccumulation in 2+ to drive catalytic H2 production. This motivated us to search for structurally modified derivatives to store electrons at more negative potentials, even after protonation. Using density functional theory (DFT) calculations, we screened electron-rich substituents and selected the introduction of an oxime group in place of the carbonyl moiety in [1](PF6)2. Herein we describe the process of this DFT-guided design, the synthesis of the selected oxime-modified PS [2](PF6)2 together with its spectroscopic and (spectro)electrochemical characterization. Under irradiation in the presence of a proton source, [2] 2+ transforms into the doubly-protonated species [2H2] 2+ , which is a stronger reductant than [1H2] 2+ .

[1H2]

Results and Discussion

In an effort to retain the charge-accumulating properties of [1] 2+ while increasing the reducing power of the stored electrons, we explored the concept of adding electron-rich groups to the existing extended ligand skeleton of [1] 2+ . The synthesis of [1](PF6)2 followed a chemistry on the complex approach [53] in which we adapted procedures previously reported for the synthesis of acridine-based tetracyclic heterocycles [54] to a Ru(II) trisdiimine framework. This method lends solubility to the otherwise insoluble planar extended ligand system. Given the success of this approach in the synthesis of [1](PF6)2, we examined known substitution reactions on the tetracyclic pyridoquinolinone core [51] to choose suitable electron-rich candidates. [54] Based on this, two electron-donating groups, an oxime ([2] 2+ ) and a methyl sulfide ([3] 2+ ), were selected (Figure 1).

Redox Process

Exp. [51] (V) S1, Table S1). b Redox potential values [2] 2+ and [3] 2+ are given relative to those computed for [1] 2+ .

E° (DFT) a (V) E° (DFT) b (V) E° (DFT) b (V) [M] 2+
[1](PF6)2 was synthesized according to the previously reported chemistry on the complex approach. [51,53] The synthesis of [2](PF6)2 was subsequently carried out from [1](PF6)2 and hydroxylamine (in-situ generated from hydroxylamine hydrochloride) in a mixed organic/aqueous solvent mixture (Scheme 1) in good yield. This reaction must be carried out in the dark, owing to the photoreactive nature of [1] 2+ in solution in the presence of electron donors (such as hydroxylamine). [51] Scheme 1. Synthesis of [2](PF6)2 from [1](PF6)2.

The planarity of the extended π-conjugated ligand in [2] 2+ promotes the formation of π-stacked dimers (see Scheme S2) of the complex in solution, as previously observed for [1] 2+ and related complexes. [46,51,53,55] This tendency to aggregate complicates the characterization of [2] 2+ by NMR spectroscopy; indeed, a concentration-dependent shift in the 1 H NMR signals of [2] 2+ (Figure S5) is observed, similar to that previously reported for [1] 2+ . In the case of [2] 2+ , however, the plot of Log(Conc. [2] 2+ ) vs. chemical shift (Figure S5) does not exhibit any plateau behavior consistent with the presence of either isolated monomers at low concentration, or isolated dimers at high concentration as further aggregation leads to insolubility. In particular, the lack of observable free monomer at the lower limit of detection strongly suggests that the formation of -stacked dimers is more energetically favorable for [2] 2+ than for [1] 2+ . In addition to the π-stacking of [2] 2+ in solution, the oxime group on the iminoquinone subunit is capable of tautomerization into the corresponding nitroso form [56] (Scheme S2). In the 1 H NMR spectrum of [2] 2+ , the merger of the signals corresponding to the two protons located on the iminoquinone moiety (H16 and H17see Scheme S3) into a single singlet at 6.05 ppm is characteristic of this additional dynamic process. Indeed, the tautomerization reaction switches the quinoidal electronic structure (oxime tautomer) to a fully aromatic ring (nitroso tautomer) (see Scheme S3), thus making the electronic environment of H16 and H17 quite dynamic. This behavior was confirmed by temperature-dependent NMR experiments (Figures S6, S7, S8 and Table S2): when the sample is cooled to 268 K (Figure S7), the singlet at 6.05 begins to separate into two broad signals, which appear to correspond to H16 and H17, although the signals fail to resolve further at lower temperatures. The complex nature of the solution-phase structure of [2] 2+ led us to the use of DFT-computed screening constants to aid in the signal assignment of the 1 H and 13 C NMR spectra (see Table S3, Figures S9 andS10). This DFT-assisted assignment demonstrates that, while dynamic processes are occurring in solution, the predominant species is the oxime form of [2](PF6)2. In order to best characterize its electronic properties, the following electronic spectroscopy and electrochemical experiments were carried out at the lowest possible concentrations permitted experimentally.

The UV/vis absorption spectrum of [2] 2+ was recorded in acetonitrile (Figure S11). In addition to the intense ligand-centered transition (bpy ligands) at 280 nm, it features a broad band in the visible region, tailing at 600 nm, thus slightly red-shifted with respect to [1] 2+ . According to our previous studies, metal-to-ligand-charge-transfer (MLCT) and ligand-centered - * transitions (on the -extended ligand) both contribute to this relatively intense absorption (molar absorptivity of 25 000 L.mol -1 .cm -1 ). [50,51] Cyclic (CV) and differential pulse (DPV) voltammetries were used to investigate the electrochemical properties of [2](PF6)2 (Figure 2). In Figure 2a, the first reduction of [2] 2+ is compared to that of [1] 2+ ; it is apparent that the first reduction of [2] 2+ occurs at a substantially lower potential than that of [1] 2+ . According to the DPV results (Figure 2c), this reduction occurs at -1.11 V vs. Fc + /Fc in [2] 2+ compared to -0.87 V vs. Fc + /Fc for [1] 2+ ; this -0.24 V experimental shift is in excellent agreement with our DFT calculations (Table 1) which predicted a decrease in reduction potential of -0.27 V. The second reduction of [2] 2+ occurs at -1.46 V vs. Fc + /Fc according to DPV, thus shifted by -0.14 V compared to second reduction of [1] 2+ (-1.32 V vs.

Fc + /Fc); again, this result matches with the DFT-predicted shift for the second reduction process (-0.17 V) (Table 1). The presence of an additional reduction process of very low intensity is also observed at -1.31 V (before the second reduction) in the DPV and is tentatively assigned to the presence of - stacked dimer; increased propensity for stacking upon reduction was indeed previously reported [46] and supported by DFT calculations. [51] -1. At potentials lower than -1.6 V vs. Fc + /Fc (Figure 2b and2c), the electrochemical behavior is similar to the previously described one for [1](PF6)2, in agreement with reduction processes occurring on the diimine ligands. We noticed the presence of an additional reduction process at -2.2 V, also tentatively assigned to the presence of aggregated species. It is important to highlight here that the first two reductions, which are of primary interest in view of photoaccumulation processes, occur at a lower potential than the previously reported system [1](PF6)2, as predicted by the DFT calculations.

In order for a complex to store multiple electrons under light-driven conditions, two important criteria must be met: the intermediate reduced states should be stable, and capable of absorbing visible light to drive an additional reduction process. To test these criteria, the (electro)chemicallyreduced forms of [2] 2+ were investigated by UV/vis absorption spectroscopy. The oxime complex was first reduced using cobaltocene (CoCp2) under strictly inert conditions, at low concentration (0.02 mM) in order to minimize the effect of π-stacking on the results. Addition of up to one equivalent of CoCp2 generated the singly-reduced [2] + species, characterized by the growth of a peak at 500 nm with a shoulder at approximately 635 nm (Figure 3a). This behavior resembles that of [1] 2+ which also developed low energy absorption peaks upon single reduction. However, unlike in the chemical reduction of [1] 2+ , [51] there is no further change in the spectrum of [2] + upon addition of a second equivalent of cobaltocene. The potential of the CoCp2 + /CoCp2 redox couple is reported to be ca. -1.3 V vs. Fc + /Fc (in dichloromethane), [57] thus not reducing enough to generate the doubly-reduced species [2] 0 as the second reduction occurs at -1.46 V vs. Fc + /Fc according to our DPV experiments.

In addition to chemical reduction, spectroelectrochemical measurements were performed to address the first and second reduction processes (Figure 3b andS12). A first controlled potential electrolysis was performed on [2] 2+ at -0.75 V (vs. Ag/AgCl); (i.e. a potential more negative than that of the [2] 2+ to [2] + reduction, see inset Figure 3b) and the evolution of the UV/vis absorption spectrum was monitored over 30 min (Figure 3b). The growth of a low energy absorption band centered at 500 nm with a shoulder at 635 nm is observed, matching the spectrum obtained by chemical reduction of [2] 2+ . The spectroscopic properties of singly-reduced [2] + are thus not affected by the higher concentration (0.25 mM in DMF) employed for the SEC measurements nor by the presence of the electrolyte salt (0.1 M nBu4PF6). The reversibility of the [2] 2+ /[2] + redox couple was assessed by poising the working electrode at +0.1 V vs. Ag/AgCl for 20 min; the resulting spectrum perfectly overlaps with the original one for [2] 2+ (Figure S12a), in agreement with a full reoxidation process and attesting for the stability of the electrochemically-generated [2] + species. In both the chemical and electrochemical reduction experiments no decrease of the absorption is observed between 400 and 500 nm upon reduction (Figure 3); the light-harvesting efficiency of the system is fully retained in the visible region to generate a reactive excited state.

A second controlled potential electrolysis was performed at -1.0 V (vs. Ag/AgCl); (i.e. a potential more negative than that of the [2] + to [2] 0 reduction, see inset Figure 3b) and the evolution of the UV/vis absorption spectrum was monitored over 30 min (Figure 3b). At -1.0 V, the absorbance spectrum tails to 800 nm and the features at 500 and 635 nm broaden when compared to the spectra collected at -0.7 V (Figure 3b). The reversibility of the reduction was again assessed by the application of a +0.1 V vs. Ag/AgCl bias for 15 minutes; when corrected for solvent loss over the course of the experiment the final re-oxidized spectrum matches the initial absorbance of [2] 2+ well (Figure S12b). The accessibility and reversibility of the second reduction by applied electrochemical potential confirms the fact that CoCp2 did not have sufficient potential to reduce The electronic structure of singly-reduced [2] + was further investigated by X-band electron paramagnetic resonance (EPR) spectroscopy, using a slightly modified version of the procedure we previously developed to study [1] + . [51] The EPR spectrum of [2] + recorded at room temperature 4 black line) features an intense radical signal with an average g value of ~2.003 and a peak-to-peak line width of 0.5 mT, characteristic of an organic radical species (consistent with the signal recorded for [1] + [51] ). The EPR signal for [2] + displays poorly resolved hyperfine splitting compared to the spectrum recorded for [1] + , which exhibits well resolved splitting (Figure S13). Such unresolved hyperfine structure can be explained by the presence of additional hyperfine coupling coming from the oxime group (one more hydrogen atom and one more nitrogen atom) relative to the carbonyl functionality in [1] + , and could indeed be simulated (Figure 4a, red line) as a monoradical coupled to 4 nitrogen and 7 hydrogen atoms (see Table S4 to form -stacked dimers compared to [1] 2+ , as previously observed in the concentration-dependent 1 H NMR experiments (Figure S5). This behavior is also consistent with previously reported studies from MacDonnell et al.. [29,46,47] Whether this spin-pairing process can result in the formation of a formal C-C bond is however still a matter for debate. [29] Electrochemically-generated [2] 0 also proved to be EPR-active; its spectrum recorded at room temperature in the electrolyte solution (0.1 M nBu4PF6 in DMF) displays five equidistant broad lines with 1:2:3:2:1 relative intensity, typical for hyperfine coupling coming from an interaction with two nearly equivalents nitrogen with a coupling constant of approximately 13.8 MHz (Figure 4b). This spectral shape is consistent with that previously reported for the singly-reduced dipyridophenazine complex [Ru(bpy)2(dppz)](PF6)2. [51,58] A similar unexpected result was previously obtained for [1] 0 [51] and might be rationalized by considering the formation of a - stacked dimer -{[2]2} 0 : in such a structure, the two electrons located on the peripheral pyridoquinolinone-oxime subunits (from the first reduction process) would be spin-paired, and the additional electrons (from the second reduction process) would reside primarily on the phenazine moiety and would be too far from each other to interact within the dimer, thus conferring its paramagnetic character. [51] Two different pathways could account for its formation: in the first one, the second reduction step would take place first to yield the doubly-reduced complex [2] 0 in a triplet spin state, followed by its dimerization to form -{[2]2} 0 . However, any attempts to detect a characteristic EPR signature of the paramagnetic triplet state of [2] 0 proved unsuccessful, though it is calculated to be slightly more stable (by 0.2 eV in vacuo) than the corresponding diamagnetic singlet state (Figure S2), in line with what was previously calculated for [1] 0 . [51] Alternatively, dimerization could occur immediately after the first reduction process to form the spin-paired dimer -{[2]2} 2+ , followed by the reduction of the phenazine moieties within this dimer to yield -{[2]2} 0 .

-0.3 V -0.75 V -1.0 V b (Figure
For both pathways, a contribution from - stacked dimers -{[2]2} + (formally [2] + + [2] 0 ), in which only one of the two phenazine rings is reduced, cannot be excluded. It should also be emphasized that, taking into consideration the relatively low intensity of the recorded signal (which represents ≈ 4% of the expected spin concentration), the formation of a high percentage of diamagnetic species (either as monomers or dimers in solution) is plausible to consider. Knowing that the prerequisites have been met for [2] 2+ to accumulate two electrons, a photolysis experiment was carried out at 0.015 mM in CH3CN, in the presence of triethylamine as the sacrificial electron donor (Figure 5). Upon visible light irradiation, a decrease in the intensity of the low energy transitions is observed together with the growth of a low energy absorbance tailing to just over 600 nm. After 10 minutes of irradiation, no further spectral change was observed. The final spectrum of the photolysis experiment is not superimposable to the ones recorded either for the singly-reduced species [2] + or for the doubly-reduced species [2] 0 (Figure 3). In the case of [1] 2+ , protons released by the decomposition of the TEA radical cation [59] were shown to protonate the reduced derivatives to promote a two-electron-two-proton redox process forming [1H2](PF6)2 rather than [1] 0 . [51] Thus, in order to identify the novel species formed by photolysis, chemical reduction of [2] 2+ was performed in the presence of trifluoroacetic acid (TFA) as a proton source. UV/vis absorption spectra were recorded after sequential additions of CoCp2 and TFA to a solution of [2](PF6)2 under strictly inert conditions (Figure S14). By contrast with the chemical reduction previously undertaken in the absence of a proton source (Figure 3), spectral changes are now observed upon the addition of a second equivalent of CoCp2. This suggests that chargecompensation through protonation occurs, forming the singly-reduced singly-protonated species [2H] 2+ [START_REF]2H] 2+ species is calculated to be thermodynamically favored and is supported by the analysis and fit of the UV-visible absorption data[END_REF] whose potential is sufficiently shifted to more positive values, with respect to [2] + , to thermodynamically allow the second reduction process. After addition of 2 eq. CoCp2 + 2 eq. TFA, the resulting spectrum overlaps very well with the normalized absorbance spectrum of the photolysis product, thus establishing that two electrons and two protons are stored on [2] 2+ under visible-light driven conditions by proton-coupled electron transfer (PCET) processes. The most stable structure for [2H2] 2+ was calculated to be the p-amino-phenylhydroxylamine derivative (Figure 6) supporting a quinone/dihydroquinone-type reduction, as previously reported for [1] 2+ .

Upon exposure to air in a dark environment, the initial spectrum of [2] 2+ was recovered (Figure S15), clearly demonstrating that the photoaccumulation process with the oxime complex [2](PF6)2 is also fully reversible. S5), in order to estimate the impact of these protonation steps on the reducing power of the charge photoaccumulation system. Charge compensation through protonation is indeed wellknown to stabilize the reduced species, although at the detriment of their reducing strength. For the first reduction process, protonation induces an anodic shift of +730 mV and +530 mV, for

[1H] 3+ → [1H] 2+ compared to [1] 2+ → [1] + , and [2H] 3+ → [2H] 2+ compared to [2] 2+ → [2] + , respectively.
Charge compensation by protonation has an even more pronounced effect on the second reduction process with shifts of +950 mV and +990 mV, for + , respectively. When the two complexes are compared, the computed redox potentials are systematically shifted by -200 to -300 mV to more negative potentials for [2] 2+ compared to [1] 2+ , demonstrating that the initially calculated trend (Table 1) is maintained in the presence of coupled protonation steps. This confirms the ability of the oxime complex [2] 2+ to store electrons, under visible light driven conditions, at more negative potentials than the parent complex [1] 2+ .

[1H2] 3+ → [1H2] 2+ compared to [1H] 2+ → [1H] + , and [2H2] 3+ → [2H2] 2+ compared to [2H] 2+ → [2H]

Conclusion

In conclusion, we have presented the DFT-guided modification of a charge-accumulating Ru(II) tris-diimine complex. The target Ru(II) pyridoquinolinone-oxime complex [2](PF6)2 features substantially decreased first and second reduction potentials, shifted by -0.24 V and -0.14 V respectively compared to the parent compound [1](PF6)2. This shift closely matches that predicted by theoretical methods (-0.27 and -0.17 V, respectively), thus validating our theoretical approach. Photoaccumulation of the doubly reduced species was accomplished by photolysis in the presence of an electron donor, via a two proton, two electron coupled reduction which produced [2H2](PF6)2. The effect of the electron-donating oxime substituent persisted in the proton coupled reduction processes and the accumulated electrons have a reducing potential 200 to 300 mV more negative than the former system [1H2] 2+ . Having successfully used our DFT-led approach to modify an existing photoaccumulation scaffold with electron-donating groups in order to lower the potential of multiple stored electrons, we foresee future applications of this system to drive bioinspired multi-electronic catalytic processes.

Experimental Section

General: The synthesis of [1](PF6)2 performed according to our previous reports. [51,53] All other chemicals were used as received. Dry solvents were used as purchased from Acros Organics, CoCp2 was purchased from Sigma Aldrich and stored at -20 °C in an argon-filled glovebox. 1 H and 13 C NMR spectra were recorded on a Bruker Avance 400 MHz or 500 MHz NMR spectrometer respectively. UV/vis absorption spectra for determination of the molar extinction coefficient of [2] 2+ were performed in acetonitrile solution on a Shimadzu 1800 UV/vis spectrophotometer under atmospheric conditions. All other UV/vis spectra were recorded in dry, degassed acetonitrile on a Cary 60 spectrophotometer in an argon-filled glovebox using a fiber optic cable. Accurate mass measurements (HRMS) were performed on a Bruker maXis mass spectrometer by the "Fédération de Recherche" ICOA/CBM (FR 2708) platform and elemental analysis on a Thermofisher Scientific "Flash 2000" by the "Plateforme d'analyse pour la chimie" (GDS 3648, Strasbourg).

Computational methods: All DFT calculations were performed with the ADF (Amsterdam Density Functional) code developed by E. J. Baerends and co-workers [61] using triple-zeta basis sets (no frozen core). Geometry optimizations were performed in vacuo relying on the Generalized Gradient Approximation (GGA) VBP exchange-correlation (XC) potential (VWN + BP: Vosko, Wilk & Nusair [62] + corrective terms by Becke [63] for the exchange, and Perdew [64] for the correlation) with ADF grid precision 6 throughout.

To compute redox potentials E°DFT(red/ox) relative to the Standard Hydrogen Electrode (SHE), we relied on the standard Born-Haber cycle (Scheme S1) for which the standard Gibbs free energy of a redox half reaction consists of the free energy change in the gas phase (ΔGin) and the solvation free energies (ΔGout) of the oxidized (ox) and reduced (red) species of a given redox couple [65,66] :

-nFE°DFT(red/ox)SHE = ΔGin +ΔGout + ΔGSHE (Eq. 1)

ΔGin contains in principle all contributions intrinsic to the cluster itself: i) the ionization energy of the reduced species, IE(red), which is computed as the enthalpic bonding energy (i.e. electronic internal energy) difference between the two redox partners, ΔEB = EB(ox) -EB(red), with the VBP XC potential, following Noodleman et al. [67] (for a more general treatment, see Batista et al. [68,69] );

ii) the internal nuclear energy Eint which is the sum of the zero-point energy, 3 kT (i.e. (3/2) kT for translation and (3/2) kT for rotation, therefore (1/2) kT for each degree of freedom), and a small correction term due to the vibrational partition function; iii) the entropic term -TΔS.

In the present case, as we are interested in relative redox potential values (i.e. redox potential values' differences), we won't compute the terms in ii) and iii) as they are expected to contribute only insignificantly to redox potentials' differences (i.e. a few hundredths of Volt). [70] ΔGout contains contributions from the interaction of the complex with its polarizable environment.

In the present case, these contributions have been limited to a dimethylformamide solvent outside the complex. Solvation energies Eenv have been therefore computed with the COSMO (Conductorlike Screening Model) [71][72][73] ADF module, representing the solvent as a dielectric continuum.

COSMO solvation energies behave according to a Born-like model, that is, they are proportional to Q² (the total cluster charge squared) multiplied by a factor of (1-1/). We considered w = 37

(via the ADF option: "solv name=dmfa") to mimick an average reaction field response of the environment.

Next, in Eq. S1, ΔGSHE = -4.43 eV is defined at pH=0. [74] For higher pH values, ΔGSHE increases by 0.059 eV per pH unit, so (for example) under standard physiological conditions at pH=7, ΔGSHE shifts to -4.02 eV. We will proceed below via a calibration procedure between computed (ref. SHE) and experimental (ref. Fc/Fc + ) redox potentials for the Ru complex with the dipyridophenazine (dppz)-pyridoquinolinone ligand, and the intercept value will implicitly contain all constant corrections (including the 0.4 V difference between SHE and Fc references.

Finally, F is the Faraday constant (here: 1 elementary electric charge) and n=1 for a one-electron reduction.

For the parent complex Chemical reduction with cobaltocene: [51] All chemical reduction experiments were performed in an argon-filled glovebox. UV/vis spectra were recorded in dry, degassed acetonitrile on a Cary 60 spectrophotometer using a fiber optic cable. All spectra were zeroed at 800 nm and corrected for increasing volume. The concentration of CoCp2 solution used for the titration was determined by adding 30 µL aliquots of CoCp2 solution to 3.0 ml of acetonitrile. The concentration was determined by measuring the change in absorbance at 327 nm between successive additions of CoCp2 using the known extinction coefficient for CoCp2 in ethanol of 7420 L mol -1 cm -1 . [75] Using the calculated concentration of 0.36 mM, it was determined that the first ~0.2 eq of CoCp2 added

to the dry degassed acetonitrile solution were used to quench residual oxygen. [29,51] To perform the chemical reduction titration of [2](PF6)2, aliquots of the CoCp2 solution (0.36 mM) were then added to a 2.5 ml solution of [2](PF6)2 (0.021 mM) in dry degassed acetonitrile and after each addition, the UV/vis absorbance spectrum was recorded. To probe the reversibility of the reduction in the presence of oxygen the solution was then removed from the glovebox and exposed to air (with gentle shaking) for 10 min. The headspace of the cuvette was then flushed with argon and the cuvette was replaced in the glove box and the final UV/vis absorbance spectrum was collected.

Chemical reduction of [2](PF6

)2 with cobaltocene in the presence of TFA: [51] Prior to chemical reduction in the presence of TFA, a chemical reduction titration was carried out with the above procedure to determine the exact volume of 1 equivalent of CoCp2 stock solution. Addition of TFA stock solution (1.5 mM) was carried out in 21 µL (1 eq.) increments. CoCp2 and TFA additions were carried out sequentially in fresh solutions of 12 µM [2](PF6)2 in dry degassed CH3CN to determine possible effects of the order of addition of reducing agent and proton source.

EPR spectroscopy experiments: Spectra were recorded at room temperature on a Bruker ER200D X-band CW EPR spectrometer with an ER-4104OR cavity with the following instrument parameters: microwave frequency, 9.53 GHz; microwave power, 3.17 + EPR and DFT data, [51] and optimized with using DFT-optimized lowest energy structure for [2] + . Sample preparations for EPR spectroscopy were carried out in an argon-filled glovebox using dry and degassed solvent. To record the spectrum of the singly-reduced complex, a 1 mm quartz cuvette (300 µL) was filled with a 200 μM solution of [2](PF6)2 (or + was determined by comparing the EPR signal intensity with that of a solution of (2,2,6,6-Tetramethylpiperidin-1-yl)oxyl of known concentration measured under the same conditions using double integration. [START_REF] Randolph | Quantitative Considerations in Electron Spin Resonance Studies of Biological Materials[END_REF]77] Percentage spin yields were of

Figure 1 .

 1 Figure 1. Structures of the parent [1] 2+ and DFT-investigated [2] 2+ , and [3] 2+ complexes (top); HOMO isosurfaces of the [M] + singly-reduced states (bottom). Isodensity value: 0.03 a.u. Rather than synthesize both [2] 2+ and [3] 2+ , DFT screening was used to predict which substituent would have the greater effect on the redox properties of [1] 2+ . To do this, the first five reduction potentials of [2] 2+ and [3] 2+ were calculated relative to those of [1] 2+ (Table 1, Scheme

Figure 2 .

 2 Figure 2. a) Comparison of the first reduction processes of [1] 2+ and [2] 2+ ; b) Cyclic voltammograms (CV) of [2] 2+ on increasing range of cathodic potentials; c) Normalized differential pulse voltammograms (DPV) of [1] 2+ and [2] 2+ . All voltammograms recorded on a glassy carbon electrode at a complex concentration of 0.25 mM in a dry, degassed 0.1 M nBu4NPF6 solution in N,N-dimethylformamide (DMF) (CV: scan rate of 100 mV.s -1 ; DPV: PHeight = 5 mV, PWidth = 50 ms, SHeight = -1 mV, ST = 200 ms).

[ 2 ]

 2 + to [2] 0 . These results overall demonstrate that the first and second reductions of [2] 2+ are both electrochemically reversible, and both reduced states are stable on the scale of minutes to hours.Additionally, both reductions produce changes in the UV/vis absorption spectrum of the complex consistent with those seen in [1] 2+ both spectroelectrochemically and via chemical reduction.

Figure 3 .

 3 Figure 3. a) UV/Vis absorption monitoring of the chemical reduction of [2](PF6)2 (0.02 mM in CH3CN) by cobaltocene (see SI for experimental details). b) UV/Vis spectra of [2](PF6)2 recorded under controlled potential electrolysis conditions (0.25 mM in DMF containing 0.1 M nBu4NPF6): black line: -0.3 V, red line: -0.75 V (first reduction process) and blue line: -1 V (second reduction process), all potentials vs. Ag/AgCl.

  for hyperfine coupling constants). The recorded EPR signals were quantified (see procedure in the experimental section) and represent, for [1] + and [2] + respectively, only ~50% and ~15% of the expected amount of radical. The lack of radical signal strongly suggests the formation of spin-paired dimers -{[2]2} 2+ (thus EPR silent) at the 0.2 mM concentration employed for the EPR experiments. Such a low amount of radical for [2] + , compared to [1] + , is in agreement with the increased propensity for [2] 2+

Figure 4 .

 4 Figure 4. Experimental (black lines) and simulated (red lines) EPR spectra of (a) [2] + in CH3CN, prepared by chemical reduction of [2] 2+ with one eq. of CoCp2; (b) electrochemically-generated [2] 0 in 0.1 M nBu4PF6 in DMF.

Figure 5 .

 5 Figure 5. UV-vis monitoring of the photolysis of [2](PF6)2 in the presence of triethylamine (0.015 mM [2](PF6)2 and 0.16 mM triethylamine in dry, degassed acetonitrile). a) Evolution of UV-vis absorbance spectra during photolysis; b) comparison of the UV-vis spectra of [2](PF6)2, the product after 10 minutes of irradiation in the presence of an electron donor, and the resulting spectrum following chemical reduction with 2 eq. CoCp2 in the presence of 2 eq. TFA.

  b

Figure 6 .

 6 Figure 6. Calculated (B3LYP) isosurfaces of the LUMO of the doubly-reduced doubly-protonated species formed during the photolysis experiment (left) and its corresponding structure (right).At this stage, a second set of redox potentials were computed for the different protonation

  mW; modulation amplitude, 0.1 mT; modulation frequency, 100 KHz; number of scan, 219 [2] + and 40 [1] + . Simulations were performed with the Matlab toolbox Easyspin (version 5.2.20). For [2] + , initial isotropic hyperfine couplings were based on [1]

[ 1 ]

 1 (PF6)) in acetonitrile. The chemical reduction was carried out by adding aliquots of a CoCp2 stock solution (50 mM in acetonitrile) and was monitored by UV-vis absorption. The solution was then transferred inside the glovebox into a flat cell fitted with Young valves. The spin concentration of [2] + and [1]

[ 2 ]

 2 Photolysis experiments: Sample preparation and photolysis experiments were carried out in an

Table 1

 1

	, Scheme

[2](PF6)2 was selected to be synthesized.

	why the methyl sulfide group has almost no effect on the second reduction process (calculated E 0
	of -0.03 eV, see
	The orbital isosurfaces of the reduced states (Figures 1, S2) help explaining how the oxime
	unit of [2] 2+ affects the second reduction potential of the complex ([2] + to [2] 0 ). The electron density
	on the dipyridophenazine part of the -extended ligand is increased in the HOMO of [2] + relative
	to that of [1] + (Figure 1), especially on the pyrazine ring where the second reduction process is
	expected to occur. This causes coulombic resistance to the addition of the second electron, thus
	lowering the potential of the subsequent reduction of [2] + to [2] 0 relative to the same second
	reduction of [1] + to [1] 0 . By contrast, the HOMO of [3] + is very similar to that of [1] + , explaining

Table 1

 1

  ).

	Table 1. Experimental and calculated reduction potentials for [1] 2+ (in V vs Fc + /Fc) and relative
	calculated potential shifts in reduction for [2] 2+ and [3] 2+ .
	[1] x	[2] x	[3] x

  (12 mL) in a schlenk tube in the dark under ambient atmosphere. The mixture was heated to 60 °C for 72 hours. The reaction mixture was quenched in the dark with a saturated solution of KPF6 (aq) (12 mL). The resulting solid was collected by suction filtration in low light and washed well with water and the crude product was passed through the fritted glass funnel with acetonitrile. The solvent was removed in vacuo, the product was dissolved in a minimum of acetonitrile and precipitated with diethyl ether. The product was collected by suction filtration, washed well with diethyl ether and dried on high vacuum to yield the pure product as a

	solution of tetrabutylammonium hexafluorophosphate (nBu4NPF6) in dry, degassed DMF under
	argon. The concentration of [2](PF6)2 was 0.3 mM. The counter electrode was a platinum wire, and
	the working electrode was a 3 mm glassy carbon disc. All voltammograms were performed using
	a saturated Ag/AgCl pseudo-reference electrode and internally referenced to the
	ferrocene/ferrocenium (Fc + /Fc) redox couple. Cyclic voltammograms were recorded at a scan rate
	of 100 mV s -1 . Differential pulse voltammograms were recorded in a negative sweep direction with
	[1](PF6)2, a plot of the experimental redox potential values (ref. Fc/Fc + ) against DFT computed values (ref. SHE) yields: E°(exp. vs. Fc + /Fc) = 1.0811  E°(DFT vs. SHE) -0.609 (Eq.2) 1 H and 13 C screening constants for both oxime and nitrosyl tautomers have been computed by DFT (ADF code) via its NMR sub-module (same technical DFT details as for the computation of redox potentials above). Synthesis of [2](PF6)2: [1](PF6)2 (50 mg, 0.044 mmol), hydroxylamine hydrochloride (77 mg, 0.92 mmol), and pyridine (72 µL, 0.89 mmol) were dissolved in a 2:2:1 mixture of without evidence of photodegradation. For elemental analysis, additional purification was KNO3 in deionized H2O) in CH3CN) and anion exchange from [2](NO3)2 to [2](PF6)2 was performed by precipitation with saturated KPF6. 1 H NMR (0.43 mM 338 K, 400 MHz, δ, ppm) 10.12 (d, J = 8.3 Hz, 1H), 9.81 (d, J = 8.2 Hz, 1.4 H), 9.73 (d, J = 9.2 Hz, 1.2 H), 9.56 J = 4.4 Hz, 1H), 8.73 (d, J = 9.3 Hz, 0.9 H), 8.56 (t, br, J = 10.8 Hz, 6.8 H), 8.25 (t, J = 4.2 Hz, 4.1 H), 8.15 (t, J = 7.9 Hz, 3.3 H), 8.03 (t, br, J = 6.5 Hz, 3.3 H), 7.98, (sept, J = 2.8 Hz, 3.4 H), 7.90 (d, br, J = 5.6 Hz, 3.3 H), 7.80 (q, br, J = 4.6 Hz, 3.1 H), 7.50 (t, J = 6.70 Hz, 3.0 H), 7.29 (quart, J = 6.1 Hz, 3.1 H), 6.82-6.95 (m, 1.2H), 6.17 (s, 0.8H). 13 C NMR (3.5 mM, 318 K, 126 MHz, δ, ppm) 179.9, 157.3, 157.1, 156.4, 154.1, 153.9, 153.6, 152.6, 152.0, 150.6, 150.3, 146.5, 145.6, 143.0, 141.7, 140.3, 139.6, 138.1, 138.0, 135.4, 134.4, 133.7, 130.8, 130.3, 130.0, 127.7, 127.4, 124.4, 124.3, 120.0, 101.8. HRMS (m/z): (M 2+ ) calc (C47H29N11ORu): 432.579453 found: 432.5797. Elemental Analysis calc (C47H29N11ORuP2F12 + 0.5 H2O + 0.3 KPF6): C 46.76% H 2.50% N 12.76% Found: C 46.89%, H 2.75%, N 12.90%. (Spectro)electrochemistry experiments: Cyclic and differential pulse voltammetry measurements were performed using a Metrohm Autolab PGSTAT100N potentiostat in a 0.1 M a step size of -0.005 V, a modulation amplitude of 0.025 V, modulation time 0.05 s, and an interval time of 0.5 s. acetonitrile:ethanol:water red-orange solid (32 mg, 0.028 mmol, 63% yield), subsequently stored in clear glass containers performed by column chromatography on alumina (from 10% to 40% of aq. KNO3 (10% v/v Sat. Spectroelectrochemical measurements were carried out on a 0.25 mM solution of [2](PF6)2 in 0.1

M nBu4PF6 in a 3 electrode cell with a 1 mm path length. The working electrode was a Pt mesh electrode, the counter electrode was a Pt wire and the reference electrode was a saturated Ag/AgCl electrode. A bias of -0.3 V vs. Ag/AgCl was applied and UV/vis absorbance spectra were recorded over 5 min with no change observed. The potential was decreased to -0.75 V (or -1.0 V as indicated) vs. Ag/AgCl for 30 min, with no further evolution observed after approximately 15 min. Finally, the voltage was increased to +0.1 V vs. Ag/AgCl for 20 min leading to the complete return to the original [2] 2+ spectrum.

Acknowledgements

Sebastian Bold is acknowledged for his help in fitting the UV/vis absorption data. This work was supported by the Labex program ARCANE (ANR-11-LABX-0003-01), the graduate school in Chemistry, Biology and Health of Univ. Grenoble Alpes (CBH-EUR-GS, ANR-17-EURE-0003) and the CEA DRF Impulsion 2017 program.


Emiliano Perez Ipiña

Stefan Otte

Rodolphe Pontier-Bres

Dorota Czerucka

Fernando Peruani
email: peruani@unice.fr

Bacteria display optimal transport near surfaces Bacteria as intermittent active chiral particles: trapped by hydrodynamics, escaping by adhesion

The near-surface swimming patterns of bacteria are strongly determined by the hydrodynamic interactions between bacteria and the surface, which trap bacteria in smooth circular trajectories that lead to inefficient surface exploration. Here, we show by combining experiments and a datadriven mathematical model that surface exploration of enterohemorrhagic Escherichia coli (EHEC) -a pathogenic strain of E. coli causing serious illnesses such as bloody diarrhea -results from a complex interplay between motility and transient surface adhesion events. These events allow EHEC to break the smooth circular trajectories and regulate their transport properties by the use stop-adhesion events that lead to a characteristic intermittent motion on surfaces. We find that the experimentally measured frequency of stop-adhesion events in EHEC is located at the value predicted by the developed mathematical model that maximizes bacterial surface diffusivity. We indicate that these results and the developed model apply to other bacterial strains on different surfaces, which suggests that swimming bacteria use transient adhesion to regulate surface motion.

The swimming of E. coli has been the focus of a great deal of research that set the basis of the canonical picture of peritrichous bacterial motion in the bulk [1][START_REF] Berg | Random Walks in Biology[END_REF][START_REF] Berg | coli in Motion[END_REF]. Bulk exploration is performed by alternating periods during which bacteria "run" in roughly straight paths and "tumbling" events that lead to abrupt changes in the moving direction. In the classical picture of bulk exploration of E. coli [1][START_REF] Berg | Random Walks in Biology[END_REF][START_REF] Berg | coli in Motion[END_REF][START_REF] Weis | [END_REF], both, the distribution of run times and durations of tumbling events are exponentially distributed. Biased motion towards chemoattractant sources has been well reported in E. coli and it has been found to rely on the capacity of bacteria of altering its moving direction by regulating the frequency of tumbling events, in particular by decreasing the tumbling frequency when heading in the direction of the chemoattractant gradient [5][6][7][8].

Surface exploration of individual bacteria, on the other hand, remains comparatively less understood. Let us stress that individual surface exploration should not be confused with colony expansion, commonly called swarming, which is a collective phenomenon typically studied at air-agar interfaces [9][10][11]. As peritrichous swimmers such as E. coli move from the bulk towards a surface, surface-induced hydrodynamic interactions impose a series of physical constraints: a) bacteria get attracted towards the surface, with bacterial motion becoming effectively two-dimensional for long periods of time [12][13][14][15][16][17], b) abrupt changes in the moving direction are strongly suppressed [18,19], and c) a hydrodynamic-induced torque (together with above-mentioned surface attraction) traps bacteria in circular, almost deterministic, trajectories [20][21][22][23][24][25] that lead to inefficient surface exploration. For a very recent and detailed study on near-surface swimming of E. coli, see [26].

Here, we show that in experiments with highly virulent, highly adhesive enterohemorrhagic E. coli (EHEC) surface exploration is characterized by the intermittency between run and stop phases, with the later involving surface adhesion. These stop-adhesion events allow EHEC to escape from the above-mentioned circular trajectories. Note that for these bacteria, surface adhesion is key for host-tissue invasion [27] and biofilm formation [28][29][30][31].

Our study reveals that surface adhesion also plays a crucial role in surface exploration. More specifically, we show that the temporal statistics of the stop and run phases is consistent with a three-state model, with two of those states connected to stop phases and at least one of them involving surface adhesion. Based on this three-state model, we build a spatiotemporal motility model. The analysis of this model reveals that the experimentally measured frequency at which stops occur is located at the theoretically predicted optimal frequency that maximizes bacterial diffusivity. We discuss the consequences of this remarkable finding. Its bimodal nature allows using the local minimum as a threshold, V /V r = 0.35. f, Distribution of changes in the moving direction (between stop and run phases), see text, p ∆θ (∆θ). g, Survivals curves Q a (τ a > t) with a = R (red squares) and a = S (blue circles); the solid curves correspond to the theoretical prediction given by Eq. ( 2) and Eq. (3). See SI for more details and movies.

I. EXPERIMENTS

We study the behavior of EHEC -strain EDL931, serotype O157:H7, which contains a large number of adhesins, including Type 1 fimbria [32] -near the bottom surface of an invitrogen Attoflour R chamber filled with 4 mm height liquid film of DMEM medium at 37 • C.

Using phase-contrast microscopy at 40x magnification, we record the behavior of bacteria near the glass coverslip at 30 frames per second. Figs. 1a-c show examples of cell trajectories (see Movie S1, S2, and S3). Bacteria display smooth, typically circular trajectories that are interrupted by stop events (indicated by red triangles) that lead to abrupt changes in the speed V (Fig. 1d) and moving direction (θ), see 1b-c and Movie S4. Note that we express the velocity vector as V (t)ê(θ(t)), with ê(•) = (cos(•), sin(•)) a unit vector. The speed distribution exhibits a bimodal shape, as shown in Fig. 1e. The local minimum displayed by the speed distribution is located at V /V r ≈ 0.35, where V r corresponds to the speed at which a local maximum is observed in the distribution (with an average V r ≈ 26µm/s). We make use of the bimodal nature of the distribution to define stop phases as those periods where V /V r ≤ 0.35 and run phases when V /V r > 0.35. This allows us to define the changes in moving direction ∆θ as result of a stop phase as the difference between the value of θ immediately before and immediately after the stopping phase. The distribution p ∆θ (∆θ), shown in Fig. 1f, is centered around 0 and p ∆θ (∆θ) > 0 in the whole interval ∆θ ∈ [-π, π].

The distribution of times associated to run (τ R ) and stop (τ S ) phases is presented in Fig. 1g in the form of survival curves Q a (τ a > t), with a = {R, S}, i.e. the probability of observing for instance a running time τ greater than t. These quantities play a fundamental role in our analysis. We find that Q R (τ R > t) is consistent with an exponential decay with characteristic time of τ R ≈ 4.4s, which lets us define the stop frequency as 1/ τ R . While the statistics of Q R (τ R > t) suggests a Poisson process regulating the run-time duration, a scenario qualitatively consistent with bulk motion [1,[START_REF] Berg | coli in Motion[END_REF], the survival curve Q S (τ S > t)

indicates that the stop phase is more complex. In particular, we find that Q S (τ S > t) is of the form A exp[-a t] + B exp[-b t] with an average stop duration τ S ≈ 0.49 s, which is about 63 % larger than the -strain-dependent -average duration of a tumbling event for non-EH E. coli, which is ≈ 0.3 s and whose distribution is a single exponential [33].

II. EXPERIMENTAL EVIDENCE OF THREE BEHAVIORAL STATES

The bimodal speed distribution, Fig. 1e, reflects an intermittent dynamics in V (t), where the speed is either fluctuating around V r or around zero, Fig. 1d. The dichotomic dynamics displayed by V (t) may lead us to think that the observed phenomenon is consistent with two behavioral states, one associated with V ≈ V r and another one with V ≈ 0. However, the fact that p S (τ S > t) is the sum of two exponentials suggests the existence of two behavioral states associated with the stop phase, and thus to the presence of a minimum of three transition from a running phase (state 2) to a stop phase not involving tethered-like motion (state 0) that transitions to a tethered state (state 1) to finally switch again to a running phase. The bacterium trajectory is shown by the red curve. See Movies S5 and S6. b, It shows that tetheredlike motion can be identified by looking at the curvature κ of the trajectory that is of the order of 1/ , with the length of the bacterium. c, It shows the survival curves p i (τ i > t) associated to state i. d, The conditions probabilities q i j that the bacterium transitions from state i to j, see text. e, The resulting Markov chain model that describes that describes the evolution of the bacterium's state. Transition rates k i j are estimated from c and d as explained in the text. For transition rate values, see text.

behavioral states, two associated to the stop phase and one to the run phase. Further inspection of the data reveals that during periods when V (t)/V r ≤ 0.35 it is often observed that the bacterium is not at rest but rotating around one of the tips of the bacterium's body: a motion that is analogous to the rotations observed in experiments with tethered bacteria [START_REF] Berg | coli in Motion[END_REF]34], providing clear evidence that these stops events involve adhesion to the surface, see Fig. 2a andb and SI for movies. This type of tethered-like motion can be easily detected by monitoring the local curvature κ(t) = 1/R(t) of the bacterial trajectory. When the trajectory corresponds to circles or circular arcs of a radius comparable to the length ( ) of the bacterium -in practice we use 1/3 < R/ < 2 -the bacterium is in a tethered state.

Tethered motion is typically not continuous, but intermittent: periods of rotational motion and no motion alternate each other. The detaching of the flagellar bundle from the surface is evidenced when the bacterium resumes free swimming and V (t)/V r > 0.35. Thus, we define the following states: "0" indicates that the bacterium is not moving -i.e. V (t)/V r ≤ 0.35 -and in a non-tethered state, "1" corresponds to the bacterium in a tethered state, and "2" implies that the bacterium is running, V (t)/V r > 0.35. Provided these definition, we can estimate from the experiments the probability of finding the bacterium in state 0, 1, or 2, denoted p i with i ∈ {0, 1, 2}. Fig. 2c shows that the corresponding survival curves are exponential. Furthermore, we can compute from the data the conditional probabilities q i j for a particle in state i to transitions to state j. Since from i, there are two possible transitions, e.g. j and k, then q i j + q i k = 1. Thus, from Fig. 2c and d all transitions rates k i j from state i to j can be estimated as follows. Knowing the average time in state i, 2c), and q i j and q i j (Fig. 2d), the rates are computed as 2019), this expression was reported with a typo: the "-1" exponent was missing]]. We find that 97% of transitions from 2 occur to state 0, and thus for simplicity in the following we neglect transitions 2 → 1. In addition, since the only detectable transition out of state 1 -i.e. involving surface detachment is by observing that the bacterium resumes swimming, i.e. transitions to 2 -the transition 1 → 0 is not present. Following the above described procedure, we obtain: k 01 = (0.20±0.02) s -1 , k 02 = (3.71±0.2) s -1 , k 12 = (0.21±0.02) s -1 , and k 20 = 0.22 ± 0.05 s -1 . Note that it is possible to conceive alternative statistical treatments that involve 6 transition rates that despite the added complexity lead to the same results, see SI. In summary, we find that the observed behavior can be accurately described by the simple Markov chain shown in Fig. 2e, whose master equation reads:

which is T i = [k i j + k i k ] -1 (Fig.
k i j = q i j T -1 i [[Note that in original publication, Perez-Ipiña et al. Nature Physics 15 610- 615 (
∂ t p 0 (t) = -(k 02 + k 01 )p 0 + k 20 p 2 , ( 1a
)
∂ t p 1 (t) = -k 12 p 1 + k 01 p 0 , (1b)
∂ t p 2 (t) = -k 20 p 2 + k 02 p 0 + k 12 p 1 . (1c)
Computing the survival curves Q R (τ R > t) and Q S (τ S > t) from these equations imply to solve first passage time probabilities on the Markov chain shown in Fig. 2e using suitable boundary and initial conditions; for details see SI. We find that

Q R (τ R > t) = e -k 20 t , (2)
while Q S we obtain:

Q S (τ S > t) = 1 - k 01 α e -(k 01 +k 02 )t + k 01 α e -k 12 t (3)
where α = k 01 + k 02 -k 12 .

III. SPATIAL MOTION

The proposed three-state model can be used as the basis for a spatial model, which should be able to describe the large variety of trajectories observed in experiments and incorporate the following considerations:

(i) Figs. 1a andb show that a moving bacterium tends to describe circular trajectories interrupted by stop events. This implies that state 2 should describe an active chiral swimmer [35][36][37] characterized by a speed V r and a radius of curvature V r /Ω 0 . The value of Ω 0 , associated to the curvature in the running state, as well as the strength of moving direction fluctuations, denoted by D θ , can be directly extracted from individual trajectories by studying the velocity autocorrelation. Further details on this procedure are provided in SI. Thus, we assigned to states 0, 1, 2 the linear speeds

V [0] = V [1] = 0 and V [2] = V r > 0,
and angular speeds Ω[0] = Ω[1] = 0 and Ω [START_REF] Berg | Random Walks in Biology[END_REF] = Ω 0 , respectively. Transitions between either stop state, i.e. 0 or 1, to state 2 involve changes in the moving direction (∆θ) as indicated in Figs. 1b andc.

(ii) We encode the details on the associated changes in the moving direction due to transitions 0 → 2 and 1 → 2 into the probability densities g 02 (θ, θ ) and g 12 (θ, θ ), respectively, where θ denotes the moving direction before the stop event and θ after it, with ∆θ = θ -θ .

Under these considerations, the spatiotemporal evolution of the three-state bacterium can be expressed via the probability density p i ( x, θ, t) of finding the bacterium in state i (with i ∈ {0, 1, 2}) at position x and moving in direction θ at time t by the following Eq. [7] simulations Eq. [7] Eq.

continuum-time forward Kolmogorov equation with transition-jumps:

∂ t p 0 ( x,θ, t) = -(k 02 + k 01 )p 0 + k 20 p 2 , (4a)
∂ t p 1 ( x,θ, t) = -k 12 p 1 + k 01 p 0 , (4b)
∂ t p 2 ( x,θ, t) = -V r ∇(ê(θ)p 2 )-Ω 0 ∂ θ p 2 +D θ ∂ θθ p 2 -k 20 p 2
+k 02 g 02 (θ, θ )p 0 dθ +k 12 g 12 (θ, θ )p 1 dθ .

Note that the spatiotemporal model defined by Eqs. [4a-4c] can also be expressed by the following set of stochastic differential equations:

˙ x = V [i(t)]ê(θ(t)) ; θ = Ω[i(t)]+ 2D[i(t)] ξ θ (t)+ζ(t) , (5)
where i(t) denotes the state of the bacterium at time t, given transition. The temporal evolution of i(t) as well as the times t k are dictated by the temporal three-state model defined above (Eq. ( 1)).

Ω[i] is defined as Ω[0] = Ω[1] = 0 and Ω[2] = Ω 0 , D[i] as D[0] = D[1] = 0 and D[2] = D θ , ξ θ (t)
In order to solve Eq. ( 4), we need to specify g 02 and g 12 . From Fig. 1f, it becomes evident that g 02 (θ, θ ) = g 02 (|∆θ|) and the same argument applies to g 12 . To simplify the calculations, we assume that g 02 and g 12 are uniformly distributed in the intervals -φ 0 ≤ ∆θ ≤ φ 0 and -φ 1 ≤ ∆θ ≤ φ 1 , respectively, and zero otherwise. Note that Eq. ( 4) can be solved without this approximation via a longer calculation that leads almost exactly the same result, see SI. We fix the values of φ 1 = π and use φ 0 to match the variance of the experimental distribution p ∆θ (∆θ), Fig. 1f, taking into account that a fraction q 02 = k 02 k 01 +k 02 of the transitions from stop to run phase occur through the path 0 → 2, while the rest (1 -q 02 ) via 0 → 1 → 2. We obtain φ 0 = 0.72π and φ 1 = π. With the above expression for g 02 and g 12 , we can analytically obtain an effective diffusion coefficient D ef f for the proposed three-state bacterial model. We use the Taylor-Kubo formula to express

D ef f = lim t→∞ 1 4t t 0 dt t 0 dt v(t )v(t )
, which reduces the complexity of the problem to compute

C(t , t ) = v(t )v(t ) = V [i(t )]ê(θ(t )V [i(t )
]ê(θ(t ) , which can be expressed as:

C(t , t ) = (6) dθ dθ i ,i V [i ]V [i ]cos(θ -θ )p(i ,θ ,t ,i ,θ ,t )
where i and i refer to the state of the bacterium at time t and t . Note that p(i ,θ ,t ,i ,θ ,t ) = p(i ,θ ,t |i ,θ ,t )p i (θ ,t ). Both, p(i ,θ ,t |i ,θ ,t ) and p i (θ ,t ) can be directly obtained from Eq. ( 4) after integrating over space Eq. ( 4). The computation of p(i ,θ ,t |i ,θ ,t ) requires solving Eq. ( 4) for t ≥ t with initial condition p k (θ, t = t ) = δ(θ -θ ) for k = i and p k (θ, t = t ) = 0 for k = i . The details on the calculation are provided in the SI.

Following the described procedure, we find that:

D eff =V 2 r D θ + k 20 φ 2 D θ + k 20 φ 2 + Ω 2 0 k st k st + k 20 , (7)
where φ = (q 0 φ 2 0 + q 1 φ 2 1 ) /6, k st is defined as k st = k 01 +k 02 1+k 01 /k 12 , and V r = 26 µm s -1 , Ω 0 = 0.3 s -1 and D θ = 0.07 s -1 are extracted from the experimental trajectories as explained in Materials and Methods and SI. The other parameters are obtained from the experimental data by the procedures explained above; values have been also provided (see previous sections). we unveil the existence of a local maximum for D eff (also global maximum). From Fig. 3b, where symbols correspond to agent-based simulations (see SI for details) and the solid curve to Eq. ( 7), it is evident that dD eff dk 20 k * 20 = 0 at an optimal k * 20 ≈ 0.2s -1 that is remarkably close to the value of k 20 obtained in experiments (see vertical line). At this value of k 20 , Eq. ( 7) predicts a diffusion coefficient of the order of 500 µm 2 s -1 , which is close to the average of experimentally obtained values, see Fig. 3d ande. On the other hand, Fig. 3c shows that D eff does not exhibit a local maximum by varying k st , which is the inverse of the average stopping time that is function of all other rates: k 01 , k 02 , and k 12 . Interestingly, the value of k st controls the potential existence of a local maximum with respect to k 20 as shown in Fig. 3f. This means that depending on the values of k 01 , k 02 , and k 12 , that may depend on the properties of bacterial adhesins on the specific surface, D eff may not exhibit a local maximum and the global maximum may be trivially located at k 20 → 0, indicating that stop-adhesion events, in this case, may not help to enhance D eff .

IV. DEPENDENCY ON THE TRANSITION RATES

V. CONCLUDING REMARKS

We have shown that in EHEC, individual surface exploration results from a complex interplay between motility and stop events leading to surface adhesion. These stop-adhesion events allow EHEC to break the hydrodynamic-induced circular trajectories and explore the surface by performing alternating run and stop-adhesion phases. More specifically, we showed that the experimental data is consistent with an arguably generic three-state model that suggests that surface exploration by swimming peritrichous bacteria can be strongly enhanced by performing transient surface adhesion events. In particular, the analysis unveiled that the average frequency at which stops occur in experiments (referred to as stop frequency) is located at the optimal value that maximizes bacterial diffusivity predicted by the developed theory. While this finding may appear as a coincidence, we found that in the experiments reported by Sauer et al. [38] with uropathogenic E. coli -which importantly possess different types of adhesins from the ones found in EHEC -and where bacteria move on and adhere to mannosylated surfaces -i.e different surfaces from the one here analyzedthe surface exploration statistics is perfectly consistent with the proposed three-state model. Moreover, the stop frequency measured in those experiments also coincides with the value that according to the developed model maximizes the diffusion coefficient of the bacteria; for details and figures on the analysis of the data reported in [38], see SI. This observation strongly suggests that the proposed mechanism is not specific to EHEC, but rather a generic mechanism of bacterial surface exploration. Moreover, it has been reported that E. coli and other bacteria possess mechano-sensitive channels that allow them to "know" when they are on a surface and to perform surface-sensing [39], as well as to regulate the activity of the bacterial flagellar motor [40]. All this suggests that bacteria may be indeed able to adapt their behavior depending on the surface properties and tune the stop frequency to the optimal value that maximizes surface diffusivity. Note that in the absence of information cues, as occurs in the studied experiments, a higher surface diffusivity confers an advantage to bacteria to find randomly distributed colonization sites or food patches located on the surface [START_REF] Adam | Stuctural Chemistry and Molecular Biology[END_REF].

In the presence of information cues, e.g. in the form of chemical gradients, bacteria have to cope with conflicting requirements of searching and localization as explained by Clark and Grant in [START_REF] Clark | [END_REF] and in the context of search without gradients in [43]. Without entering into these important issues [START_REF] Clark | [END_REF]43], the analysis of the model suggests that by regulating the frequency of stop-adhesion events bacteria can perform biased motion on surfaces (see SI).

Undoubtedly, surface motility experiments in well-controlled chemoattractant gradients are required to clarify these intriguing issues. Other promising experimental directions include the use of c-di-GMP to alter the run time distribution [44], alter the adhesion properties by using a different substrate material [45] or the use of mannose as in [46] to alter the average stop time.

The collected evidence, together with the recent discoveries on bacterial surface sensing capacity, and the observation that the experimental values of the tumbling frequency in three-dimensional swimming do not maximize bacterial 3D diffusivity [START_REF] Berg | coli in Motion[END_REF], let us speculate that peritrichous bacteria have evolved to become optimal surface explorers by the identified exploration mechanism involving transient adhesion events. This should not come as a surprise if we consider that nutrients in aqueous environments tend to accumulate at surfaces [47]. Preparation for video-microscopy: Bacteria were grown overnight into LB broth medium without shaking (condition that preserved flagellum). Bacteria were pelleted by gentle centrifugation (2500 rpm for 10 minutes) and resuspended in DMEM medium. Subsequently, 2 ml of liquid was disposed into invitrogen Attoflou R cell chambers, leading to a density of approx. 10 6 bacteria/dish. The circular cell chambers have a diameter of 25 mm, which lead to a heigth of approx. 4 mm of the liquid above the bottom glass surface of the cell chamber.

For time-lapse video microscopy, the chambers were placed in a humidity (95%), CO2 (5%) and temperature (37 o C) -controlled environment. The focus was set on the cover slip at the bottom of the cell chamber, in order to record bacterial motion close to the surface glass/liquid. At least 30 minutes were given to the system to equilibrate prior to recording bacterial motion.

Records of video-microscopy: Motile bacteria were recorded by phase-contrast microscopy using a Leica DMI6000 B inverted microscope equipped with a high-sensitive Ropper Cool-Snap HQ2 CCD camera (Photometrics) at 40x magnification (numerical aperture: 0.75, Leica HCX PL Fluotar PH2). Images were acquired with the LAS-AF software (Leica, Germany) at a rate of 35 fps.

Data analysis and fitting of parameters

Videos were analyzed using fiji platform [48]. Trajectories of 147 individual bacteria were tracked using two different tools: MTrackJ [49] for manual tracking and TrackMate [50] for semi-automatic tracking. From the tracking, the bacteria's center of mass { x j } with x j = (x(t j ), y(t j )) at times t j = j • ∆t for j = 1, ..., N time steps were obtained for each bacterium, where is ∆t = f -1 M the inverse of the microscope frame rate. The speed at time t j was computed as

V j,n = 1 n∆t || x j+ n 2 -x j-n 2 ||
i.e. as the average speed over the neighboring n = 6 time steps in order to remove rapid speed fluctuations. For each bacterium, a reference speed V r was computed as the arithmetic average of a subset of speed values {V s } that satisfy V s ≥ 0.5V max , where V max = max{V j } is the maximum speed of the trajectory. As was mentioned in the main text, V r was used to identify the stops events. Plotting the distribution of V j,n /V r for all the trajectories we observed two local maximums, one at V r and the other at 0. A minimum value is located at V j,n /V r = 0.35 and was used as a threshold to identify run and stop phases. A stop phase is defined as the consecutive times s i = {t j } where the condition V j,n /V r < 0.35 is satisfied. In the same way, a run phase corresponds to consecutive times r l = {t k } where V k,n /V r > 0.35.

The moving direction θ(t j ) was computed by first smoothing the x and y components of the trajectory independently using a Gaussian filter with width σ = 3 (in order to remove fluctuations such as wobbling motion of the bacterial body) and subsequently using the quadrant-sensitive arctangent function on the smoothed trajectory { x S j }:

θ j = arctan y S j+1 -y S j x S j+1 -x S j .
The change in the moving direction during a stop event was defined as ∆θ = θ(min(r l ))θ(min(s i )), again where the r l -event is the running event that immediately follows the s ievent.

In order to estimate the population mean values of the parameters we only consider the running phases longer than 100∆t. Then we estimated the mean value of the running speed,

V r as the average over all the V r . In the case that a trajectory did not have any stops we re-defined V r as the average in time of V j,n . The angular speed, Ω 0 , was computed using the curvature κ of the running phases. The curvature κ can be geometrically calculated by,

κ(t i ) = - | ¨ x(t i ) × ˙ x(t i )| | ˙ x(t i )| 3 = ẋ(t i )ÿ(t i ) -ẏ(t i )ẍ(t i ) ( ẋ2 (t i ) + ẏ2 (t i )) 3/2
The temporal first and second derivatives appearing in this equation are estimated using finite differences. Finally, Ω 0 and κ are geometrically related by

Ω 0 = κV r , (8)
such that Ω 0 can be obtained if previously κ and the mean speed V r have been measured

independently.

The angular diffusion coefficient, D θ , was computed from the (average) correlation of the moving direction, ˆ e i . ˆ e i+j i , where ˆ e i = ˙ x(t i )

V i,n is the (average) direction of motion of the particle at time t i . For a chiral active particle the correlation of the moving direction can be expressed as: ˆ e i . ˆ e i+j i = e -D θ j∆t cos (Ω 0 j∆t) .

Finally, knowing Ω 0 , it is possible to extract the values of D θ fitting the data with Eq. ( 9).

All fitting was done using the Levenberg-Marquardt nonlinear least-squares algorithm. In SI the histograms of the estimated values of the model parameters obtained for the running events analyzed are shown.

Notes on the time distributions estimations

For the analysis of the duration of the runs and stops we have to consider that we can track bacteria as long as they are in the camera frame, meaning that we cannot follow them for long times and we only see a part of their trajectories. This implies that we have a bias in the accessible data towards the short times. In the analysis of the duration of run and stop phases, in order to avoid a bias towards short times, "partial" events were included, i.e. those events that were not observed from the beginning till the end, e.g. the partial running event of a bacterium that enters/exits the camera frame (in such cases we do not know when the running event started/finished). These partial events were analyzed following the Kaplan-Meier method to estimate the survival distribution. For the running time distribution we applied the same procedure as the one used in [1] in order to account for the individual variability.

FIG. 1 .

 1 FIG. 1. Experimental trajectories and their statistics. a-c, EHEC display smooth circular trajectories (white curves) [a] that are interrupted by stop events (indicated by red triangles) that lead to abrupt changes in the speed (V ) and reorientations (∆θ) [b and c]. d, The characteristic intermittent behavior of the speed V (t) over time t, whose value alternates between V r and 0. Run-time (τ R ) and stop-time (τ S ) periods are indicated, see text. e, Speed distribution p v (V /V r ).

FIG. 2 .

 2 FIG. 2. Evidence of three states. a, Example of the evolution of a bacterium that undergoes a

FIG. 3 .

 3 FIG. 3. Diffusion coefficient. a, The value of D eff , color coded and measured in units of µm 2 s -1 , as function of two transition rates, k ij and k i j . The star in each panel indicates where the experimental value is located. b, D eff vs the stop frequency k 20 . Symbols corresponds to agent-based simulations, the solid black curve to Eq. (7), and the vertical red line indicates the experimentally measured value of k 20 . c, D eff vs the inverse of the average stopping time given by k st = k 01 +k 02 1+k 01 /k 12 . Symbols corresponds to agent based simulations obtained by varying k 12 (triangles), k 01 (squares), and k 02 (crosses), while the solid curve corresponds again to Eq. (7). The vertical arrow indicates experimentally obtained k st value. Note that D eff does not exhibit a local maximum of any value of k st . d, Experimentally estimated values of D eff , where the horizontal dashed lines corresponds to the average. e, It shows the histogram of the D eff values displayed in d, where the vertical red dashed line corresponds to the average D eff . f, D eff vs k 20 for k st values 0.010, 0.056, 0.316, 1.778 and 10 4 s -1 in the direction indicated by the horizontal arrow. Note that for low k st values, there is no optimal k 20 .

Fig. 3a shows

  Fig. 3a shows the value of D eff color coded as function of two rates k ij , with all other parameters kept constant to the experimentally measured values. In each panel, a star indicates the location of the experimentally obtained values of the rates. The exploration of the model let us identify the role and relevance of each rate. The most remarkable finding is that for large enough values of k 02 and k 12 , and for the whole range of k 01 , by varying k 20

  Enterohemorrhagic Escherichia coli EDL931 (CIP 103571 "Institut Pasteur") serotype O157:H7 (EC) was kindly provided by Stéphane Méresse, Faculté des Sciences de Luminy, Centre d'Immunologie de Marseille-Luminy (CIML), INSERM-CNRS, Marseille, France. Bacteria were stored in Luria-Bertani (LB) medium plus 15% glycerol at -80 • C.

ACKNOWLEDGMENTS

We thank L. Gómez Nava, R. Großmann, A. Be'er, and Giovanni and Giorgio Volpe for insightful comments on the text. Experiments were performed at C3M Imaging Core Facility (Microscopy and Imaging platform Côte d'Azur, MICA). We acknowledge support from Grant ANR-15-CE30-0002-01 and from Biocodex.