Pascal Didier
email: pascal.didier@unistra.fr

Jagannath Kuchlyan

Lara Martinez-Fernandez

Pauline Gosset

Jérémie Léonard

Yitzhak Tor

Roberto Improta

Yves Mely
email: yves.mely@unistra.fr

Yves Mély

Deciphering the pH-dependence of ground-and excited-state equilibria of thienoguanine

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

Introduction

Numerous cellular mechanisms and pathways rely on dynamic interactions of nucleic acids with proteins that induce local and transient changes in their secondary and tertiary structures as well as in their properties and functions [START_REF] Mckinney | Structural dynamics of individual Holliday junctions[END_REF][START_REF] Kaushik | A bouquet of DNA structures: Emerging diversity[END_REF][START_REF] Aznauryan | A direct view of the complex multi-pathway folding of telomeric G-quadruplexes[END_REF][START_REF] Dzatko | Evaluation of the Stability of DNA i-Motifs in the Nuclei of Living Mammalian Cells[END_REF][START_REF] Lilley | Structures of helical junctions in nucleic acids[END_REF][START_REF] Sholokh | Environmentally Sensitive Fluorescent Nucleoside Analogues for Surveying Dynamic Interconversions of Nucleic Acid Structures[END_REF]. Though X-ray diffraction, NMR spectroscopy or electron microscopy provide invaluable information on the structure of protein/nucleic acid complexes, they are less suited for monitoring the dynamics of these interactions and their associated changes, especially in dilute solutions. Due to their exquisite sensitivity, fluorescence-based techniques are highly effective for such purposes. Their implementation for nucleic acids, however, suffers from the lack of emissive nucleoside analogues able to reliably replace the natural nucleobases, while maintaining favorable photophysics. Indeed, although many structurally diverse nucleotide analogs exhibiting probe-like properties have been developed, most of them are strongly quenched upon incorporation into oligonucleotides (ODNs) and/or do not properly substitute the natural nucleobases, inducing substantial changes in the conformation and dynamics of the labeled sequences 7-10 . This is notably the case for 2-Aminopurine (2-AP), a very popular mimic of adenosine that is strongly quenched in duplexes and alters the dynamics of the flanking base pairs [START_REF] Dallmann | 2-Aminopurine Incorporation Perturbs the Dynamics and Structure of DNA[END_REF]. To overcome these limitations, efforts have been devoted to designing new generations of fluorescent nucleoside surrogates [START_REF] Xu | Fluorescent nucleobases as tools for studying DNA and RNA[END_REF][START_REF] Wilhelmsson | Fluorescent analogs of biomolecular building blocks: design and applications[END_REF]. Among these new compounds, a particularly interesting breakthrough has been recently achieved with the introduction of thienoguanosine ( th G), a truly faithful emissive and responsive G surrogate, which reliably reproduces the structural context and dynamics of the parent native nucleoside [START_REF] Shin | Emissive RNA Alphabet[END_REF]. In addition, this fluorescent G analogue remains strongly emissive when incorporated into ODNs, and exhibits environmental sensitivity. This fluorescent nucleoside has already been extensively used to monitor protein/ODN interactions [START_REF] Sholokh | Environmentally Sensitive Fluorescent Nucleoside Analogues for Surveying Dynamic Interconversions of Nucleic Acid Structures[END_REF][START_REF] Sholokh | Tautomers of a Fluorescent G Surrogate and Their Distinct Photophysics Provide Additional Information Channels[END_REF][START_REF] Han | Approach to the Investigation of Nucleosome Structure by Using the Highly Emissive Nucleobase th dG-tC FRET Pair[END_REF], protein-induced base flipping [START_REF] Kilin | Dynamics of Methylated Cytosine Flipping by UHRF1[END_REF], conformational changes in ODNs [START_REF] Sholokh | Environmentally Sensitive Fluorescent Nucleoside Analogues for Surveying Dynamic Interconversions of Nucleic Acid Structures[END_REF][START_REF] Park | Highly emissive deoxyguanosine analogue capable of direct visualization of B-Z transition[END_REF], single nucleotide polymorphism [START_REF] Sholokh | Tautomers of a Fluorescent G Surrogate and Their Distinct Photophysics Provide Additional Information Channels[END_REF]; ribozyme-mediated cleavage processes [START_REF] Li | Polymerase-Mediated Site-Specific Incorporation of a Synthetic Fluorescent Isomorphic G Surrogate into RNA[END_REF], cellular activity of siRNAs [START_REF] Shin | Cellular activity of siRNA oligonucleotides containing synthetic isomorphic nucleoside surrogates[END_REF] and distance measurements in DNA [START_REF] Han | Approach to the Investigation of Nucleosome Structure by Using the Highly Emissive Nucleobase th dG-tC FRET Pair[END_REF][START_REF] Han | Development of a Vivid FRET System Based on a Highly Emissive dG-dC Analogue Pair[END_REF].

In a recent study, we demonstrated that th G exists as two ground-state tautomers with distinct absorption and emission spectra, and high quantum yield in aqueous buffer at neutral pH [START_REF] Sholokh | Tautomers of a Fluorescent G Surrogate and Their Distinct Photophysics Provide Additional Information Channels[END_REF][START_REF] Martinez-Fernandez | Excited-State Dynamics of Thienoguanosine, an Isomorphic Highly Fluorescent Analogue of Guanosine[END_REF]. Using quantum mechanical calculations, the two tautomers identified as the H1 and H3 keto-amino tautomers (Figure 1) were predicted to coexist both in the isolated nucleoside or when included in ODNs. Moreover, these tautomers exhibit distinct environmental sensitivity and thus provide additional data channels for analyzing G residues in ODNs and their complexes with proteins.

In this context, our objective was to further characterize the photophysical properties of this information-rich fluorescent nucleobase, by performing a systematic study as a function of pH (2 to 12). Steady-state and time-resolved experiments allowed us to support the existence and characterize three new charged species, as well as the equilibria between these forms and the already characterized H1 and H3 tautomers in both the ground-and excitedstates. In addition, (TD)-DFT calculations were used to substantiate their structure and rationalize their observed spectroscopic properties. The characterization of the pH dependent complex equilibria between the different th G forms opens a pathway for new applications, such as pH sensing.

Experimental section

Materials 2-amino-3H,4H-thieno [3,4-d]pyrimidin-4-one was synthesized as reported by [START_REF] Shin | Emissive RNA Alphabet[END_REF] [START_REF] Shin | Emissive RNA Alphabet[END_REF]. This compound also called thienoguanine ( th Gb) corresponds to the nucleobase of th G and is thus not coupled to a ribose or deoxyribose sugar. th Gb was selected for the present study, because of the large quantities of compound needed and the possible pH-dependent alterations of the sugar at extreme pHs. Stock solutions of th Gb were prepared in spectroscopic grade DMSO. For measurements at the different pH values, th Gb stock solutions were diluted at a final concentration of 12 µM in aqueous solution (<0.2% DMSO). Fresh solutions were prepared for each pH value. Solutions at pH 2 and 3 were prepared using dilute HCl. Solutions at pH 4 and 5 were prepared by using sodium 25 mM acetate-acetic acid buffer. Solutions at pH 6 and 7 were made by using 25 mM MES and Tris buffer, respectively. For pH 8 and 9, 25mM HEPBS buffer was used. Finally, solutions at pH 10-12 were prepared using dilute NaOH. All buffer reagents were purchased from Sigma Aldrich.

Steady-State Spectroscopy

Absorption spectra were collected on a Cary 4000 UV-visible spectrophotometer (Varian). Fluorescence excitation and emission spectra were recorded at 20°C on a FluoroLog spectrofluorometer (Jobin Yvon) equipped with a thermostated cell compartment. The spectra were corrected for buffer fluorescence, lamp fluctuations, and detector spectral sensitivity. Excitation spectra of th Gb were collected at emission wavelengths of 375, 400, 450, 500, and 550 nm. For emission spectra, excitation wavelengths were at 310, 320, 330, 340, 350, 360, and 370 nm. Quantum yields (QYs) of the th Gb solutions were determined using quinine sulphate (QY=0.546 in 0.5 M H2SO4) as a reference [START_REF] Melhuish | QUANTUM EFFICIENCIES OF FLUORESCENCE OF ORGANIC SUBSTANCES: EFFECT OF SOLVENT AND CONCENTRATION OF THE FLUORESCENT SOLUTE 1[END_REF].

Deconvolution of the absorption and emission spectra was performed based on a linear combination of the spectra of the different pure forms of th Gb. Matlab was used to determine the contributions of the different forms by using a spectral deconvolution algorithm. The ground-state pKa values were determined by fitting the dependence of x, the relative concentrations of each species (as determined from the deconvolution of the absorption spectra) or the absorbance maxima as a function of pH with a model (1) derived from the Henderson-Hasselbalch equation, where x1,2 are the relative concentrations or absorption wavelengths measured at low and high pH, respectively.

𝑥 = # 𝑥 ! × 10 (#$%#&' ! ) + 𝑥 ) 1 + 10 (#$%#&' ! ) *+!,) (1)
The pKa* values were calculated using the Förster Cycle theory with the following equation:

𝑝𝐾 ' * = 𝑝𝐾 ' - (ℎ𝜈 ! -ℎ𝜈 ) ) 2.303𝑘 . 𝑇 (2)
where hn1 and hn2 are the energy of the 0-0 electronic transition of the acid and basic forms, respectively. The corresponding energies were estimated from the average of the absorption and emission maxima of the acid and basic forms [START_REF] Lakowicz | Principles of fluorescence spectroscopy[END_REF].

Time-resolved Spectroscopy

Time-resolved fluorescence measurements were performed using time-correlated single-photon counting technique with excitation at 315 nm and 360 nm. Excitation pulses at 315 nm were generated by a pulse-picked frequency-tripled Ti-sapphire laser (Tsunami, Spectra Physics) pumped by a Millenia X laser (Spectra Physics) with a repetition rate of 4 MHz. Excitation pulses at 360 nm were generated Please do not adjust margins Please do not adjust margins by a supercontinuum laser (NKT Photonics SuperK Extreme) with 10 MHz repetition rate. The fluorescence decays were collected throughout the emission spectra with 20 nm interval, using a polarizer set at magic angle and a 16 mm band-pass monochromator (Jobin Yvon). The single-photon events were detected with a microchannel plate photomultiplier R3809U Hamamatsu, coupled to a pulse pre-amplifier HFAC (Becker-Hickl GmbH) and recorded on a time-correlated single photon counting board SPC-130 (Becker-Hickl GmbH). The instrumental response function (IRF) recorded with a polished aluminum reflector showed a full-width at half-maximum of 40 ps for 315 nm excitation and 50 ps for 360 nm excitation.

Time-resolved exponential decays were fitted by using the global fit procedure of Igor Pro (Wavemetrics). The fitting function was a sum of exponential decays (up to 4 components) convolved with a normalized Gaussian curve of standard deviation σ standing for the temporal IRF and a Heavyside function. All emission decays were fitted using a weighting that corresponds to the standard deviation of the photon number squared root. The lifetimes were shared for all emission wavelengths while the amplitudes were kept as free parameters during the fitting (Levenberg-Marquardt algorithm). Decay Associated Spectra (DAS) were constructed by: Ii(λ)=αiτiI(λ)/∑ αi(λ)τi, where I(λ) is the steady-state emission spectrum and αi (λ) are the wavelength-dependent amplitudes.

Femtosecond Transient Absorption (TA) spectroscopy

The TA set-up is based on a chirped-pulse, regenerative amplifier (Amplitude company) operating at 5 kHz, and producing 800 nm, 40 fs, 0.5 mJ pulses used to pump a Non-collinear Optical Parametric Amplifier (TOPAS; Light Conversion) followed by second harmonic generation. [START_REF] Skilitsi | Ultrafast photophysics of the environment-sensitive 4ʹ-methoxy-3-hydroxyflavone fluorescent dye[END_REF] The resulting light beam was tuned at 315 nm or 360 nm and used as pump pulse to excite the samples. As a probe pulse, we have used a chirped, white-light supercontinuum (300-700 nm) produced by focusing a few μJ of the fundamental 800 nm pulse inside a 2 mm thick CaF2 crystal. The pump and probe are linearly polarized, in magic angle configuration, and focused into a 0.5 mmthick flow cell where the samples are circulated with a peristaltic pump. The spectrum of the transmitted probe is collected with a spectrograph and a CCD camera. The measured signal corresponds to the pump-induced absorbance ΔA as a function of pump-probe delay τ and probe wavelength λ. The experimental time resolution is 60 to 80 fs. After each TA experiment, we record the solvent signalsometimes referred to coherent artifact. This signal is used to measure the wavelength dependence of time zero -resulting from the group velocity dispersion of the probe beam [START_REF] Kovalenko | Femtosecond spectroscopy of condensed phases with chirped supercontinuum probing[END_REF]. We perform conventional TA data analysis, which consists in subtracting this solvent signal from the data recorded on the molecules in solution and post-processing the data to correct the chirp. Then, we perform the global analysis. Singular Value Decomposition (SVD) is used to decompose the data as a product of three matrices: singular spectra, singular values (SV), and singular transients (ST). A global fitting is realized on the two or three dominating ST. The number of STs considered for the fit is such that the neglected ST's should have an amplitude weaker than the residuals of the global fit. The fitting function is a sum of exponential functions convolved with a Gaussian function modeling the Instrument Response Function (IRF). Up to three time constants are needed to minimize the Chi-square of the global fit. Finally, each time constant determined by the fit, is associated to a Decay Associated Spectrum (DAS) defining the wavelength dependence of the corresponding preexponential factor and therefore the spectral evolution associated to each time constant. For these measurements, th Gb stock solutions were diluted at a final concentration of 1 mM in aqueous solution (<2% DMSO).

Computational details

Minima characterization. Ground-and excited-state energy minima of the different species were optimized by means of Density Functional Theory (DFT) and its time-dependent version (TD-DFT), using the PBE0(26,27) functional and 6-31+G(d,p) basis set. Their energies (and free energies G wa ) were computed by single point calculations using a larger basis set, 6-311+G(2d,2p). To model solvent effects, we resorted to a mixed discrete continuum model, where 6 water molecules were explicitly included in the calculations (Figure 1b) while bulk solvent effects are taken into account by the Polarizable Continuum Model (PCM) [START_REF] Miertuš | Electrostatic interaction of a solute with a continuum. A direct utilizaion of AB initio molecular potentials for the prevision of solvent effects[END_REF][START_REF] Tomasi | Quantum Mechanical Continuum Solvation Models[END_REF]. This computational approach has already profitably been used to study th G, providing an accurate description of the tautomerization equilibria and the electronic spectra [START_REF] Martinez-Fernandez | Excited-State Dynamics of Thienoguanosine, an Isomorphic Highly Fluorescent Analogue of Guanosine[END_REF]. We shall use the computed vertical absorption and emission energies to interpret the experimental spectra. Actually, these quantities cannot be directly compared with the absorption and emission maxima, which strongly depend on the vibrational overlap between the different excited states [START_REF] Ferrer | Insights for an Accurate Comparison of Computational Data to Experimental Absorption and Emission Spectra: Beyond the Vertical Transition Approximation[END_REF]. As a consequence, also considering the other possible source of errors of our calculations (density functional, basis set, treatment of solvent effect, lackingof thermal effects), discrepancies of 0.2-0.3 eV between vertical energies and spectral maxima are not surprising and cannot be expected to be the same for all the molecules. pKa and lifetimes. For both ground and excited states, pKa values were obtained using:

pKa=DG wa /(2.303 RT) (3)
DG wa corresponds to the free energy difference (in kcal/mol) between the neutral and acidic/basic forms computed in water (using the experimental value, 269 kcal/mol, for the free energy of the H + in water) [START_REF] Topol | On the structure and thermodynamics of solvated monoatomic ions using a hybrid solvation model[END_REF][START_REF] Tissandier | The Proton's Absolute Aqueous Enthalpy and Gibbs Free Energy of Solvation from Cluster-Ion Solvation Data[END_REF]. We have double-checked our pKa predictions by using also another procedure [START_REF] Saracino | Absolute pKa determination for carboxylic acids using density functional theory and the polarizable continuum model[END_REF], which combines DG computed in the gas phase at a higher level of theory with differential solvation energies (see SI).

Radiative fluorescence lifetimes were estimated using the protocol described in ( 21) that considers the emission energies (VEE) and the dipole moments (𝜇): This journal is © The Royal Society of Chemistry 20xx

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𝜏 ! = 1 𝑘 ! #$%&' 𝑘 ! = 4 3 VEE ( 𝑐 ( 𝜇 )
The fluorescence lifetime ( c t) can be estimated by considering the experimental quantum yield (Φ) and the relationship below:

𝛕 𝐜 = 𝜏 "#$%& ' 𝑥 Φ

Results

Absorption properties

The absorption spectra of th Gb were recorded in aqueous buffers in the pH range 2-12 (Figure 2A andB). In our previous work,( 14) we demonstrated that the spectrum of th G at pH=7.5 is the sum of two contributions assigned to the H1 and H3 tautomers, characterized by distinct absorption maxima (λmax) at 334 and 313 nm, respectively. The absorption spectra of H1 and H3 were obtained using the excitation spectra at 500 nm and 380 nm, where only H1 and H3 emits, respectively( 14). Here we show that for th Gb, the measured spectra and their absorption maxima (Figure 2C) remain almost superimposable within the 6-9 pH range, suggesting that the H1 and H3 tautomers are the only species in this pH range. Their equilibrium is obviously independent of pH in this range, indicating that their interconversion does not involve any protonation or deprotonation step. In addition, the measured spectra for th Gb tautomers were identical to the one already published [START_REF] Sholokh | Tautomers of a Fluorescent G Surrogate and Their Distinct Photophysics Provide Additional Information Channels[END_REF]. In contrast, a progressive blue shift is observed at lower pH values, converging towards a spectrum that is almost superimposable at pH 2 and 3. At the two latter pH values, the position of the absorption maximum (309 nm, Figure 2C) is blue-shifted in respect to the absorption maxima of the H1 and H3 forms (334 and 313 nm, respectively), suggesting that a new species, arbitrarily called the A form, is generated at these acidic pHs. In parallel, a strong red shift accompanied by an absorbance increase is observed at pH > 9, converging towards a spectrum that is almost superimposable at pH 11 and 12. At such basic pHs, the position of the absorption maximum (340 nm, Figure 2C) is significantly red-shifted as compared to the H1 and H3 spectra, indicating that an additional form, called the B form, appears at basic pH. Taken together, our absorption data suggest that in addition to the H1 and H3 tautomers, two additional forms appear at acidic and basic pH. Assuming that the acidic A and basic B forms of th Gb are the only species in solution at pH 2 and 12, respectively, we could deduce their molar extinction coefficients. Values of 4210 M -1 cm -1 at 309 nm, and 5950 M -1 cm -1 at 340 nm, were obtained for pH 2 and 12, respectively.

Knowing the individual absorption spectra of all four th Gb species (Figure 3A), we can recover the relative concentrations of the different species in the pH range 2-12 (Figure 3B). The relative concentration of the A form was found to drop at pH>3 and become negligible at pH 6. The B form was observed at pH>9, reaching 100% at pH ³ 11. The relative concentrations of the H3 and H1 tautomers were constant at pH 6-9, being about 43±1% and 57±1%, respectively. Interestingly, the concentrations of both tautomers were found to decrease at pH 4 and 5, as well as at pH 10, but their ratio remained constant. This confirms that the value of the equilibrium constant between the two tautomers, K=[H1]/[H3]=1.33±0.05, is independent of pH.

Moreover, the changes in the concentrations of both A and B forms with pH can be used to determine their pKa values using the Henderson-Hasselbalch equation (Fig 3B)

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Emission features

To confirm and further characterize the different forms of th Gb, the emission spectra of th Gb were recorded at different excitation wavelengths selected over the entire absorption spectrum. The emission spectra in the pH range 6-9 (Fig. 4E) were very similar to the previously reported spectra of th G at pH 7.5 [START_REF] Sholokh | Tautomers of a Fluorescent G Surrogate and Their Distinct Photophysics Provide Additional Information Channels[END_REF] and exhibited comparable dependence on the excitation wavelength, confirming that the equilibrium between the H1 and H3 tautomers is pH independent in this range. tautomers. At pH 10 (Fig. 4F) and excitation wavelengths ³ 350 nm, the emission maximum was at 458 nm, and thus intermediate to that of the H1 tautomer (470 nm) and the B form (451 nm). This strongly suggests that both H1 and B forms are emitting at this pH. Moreover, the H3 emission could be clearly evidenced at pH=10, when the excitation wavelength is below 350 nm. The simultaneous presence of the three forms is further confirmed from the excitation spectra ( Please do not adjust margins Please do not adjust margins emission spectra suggest that the A form contributes to the emission process. In addition, the independence of the excitation spectra on the emission wavelength suggests that only the A form absorbs and emits at pH 2 (Fig S3).

The results obtained at pH 4 and 5 were more difficult to interpret. Though the absorption spectra suggested that the ratio between the H1 and H3 concentrations was constant over the pH range 4-10, we clearly observed that the relative contribution of the emission of the H3 tautomer decreased at pH 4 and 5 (compare Fig 4C andD with Fig. 4E). Moreover, at pH 3, although the contribution of the H1 tautomer was less than 10% in the absorption spectrum, the emission spectra were found to be independent on the excitation wavelength and to fully overlap with the emission spectrum of the H1 tautomer (Fig 4B). The clear discrepancy between the absorption and emission data at pH 3-5 thus suggests a possible excited-state reaction, which accumulates the H1 tautomer.

We then characterized the fluorescence quantum yields of th Gb as a function of pH. For each pH value, we determined the QYs at multiple excitation wavelengths (310-340 nm at pH 2 and 3, and 310-370 nm at pH 4-12). At all pH values, the QY values were found to be constant over the explored excitation wavelength range. Moreover, the QY values were found to be remarkably constant (0.48+/-0.03) over the pH range 5-12 (Fig S4). As in the pH range 6-9, only the H1

tautomer is excited at wavelengths ³ 350 nm, while both H1 and H3 tautomers are excited at lower wavelengths, this clearly indicates that both H1 and H3 tautomers possess the same QY (0.48±0.03). This conclusion is in line with our previous contribution, where a QY of 0.49(±0.03) was reported for both H1 and H3 tautomers of th G at pH 7.5 [START_REF] Sholokh | Tautomers of a Fluorescent G Surrogate and Their Distinct Photophysics Provide Additional Information Channels[END_REF]. Moreover, as only the B form is observed at pH 12, while the B form is mixed with the H1 and H3 tautomers at pH 10 and 11, the constancy of the QY value at pH 10-12 strongly suggests that the QY of the B form is identical to that of the H1 and H3 tautomers. In contrast, the A form exhibits a substantially lower QY (0.11±0.01), as measured at pH 2. Knowing the percentages of H1, H3 and A forms in the ground state at pH 3-5 (Fig 3B) and the QYs of the three forms, it is easy to calculate the QYs that would be expected in this pH range if no excited state reaction would occur. The calculated values (0.13, 0.23 and 0.41 at pH 3, 4 and 5, respectively) are markedly lower than the experimental ones (0.3, 0.42, and 0.48), suggesting that the A form must be converted in one of the other forms in the excited state.

Time-resolved spectroscopy

To gather additional information on the excited-state reaction at acidic pH and the photophysics of th Gb, its time-resolved fluorescence decays were analyzed. First, we recorded spectrally resolved fluorescence decays as a function of pH (4-12) upon excitation at 360 nm. As depicted in Figure 3A, only the H1 tautomer and the B form absorb at this wavelength. For each pH value, I(l,t) was fitted using a global analysis algorithm. The lifetimes were kept constant for all pH and l values and the amplitudes were the only free parameters of the fit. The DAS were calculated as described in the materials and methods section. Within the pH range 4-9, the fluorescence decays were characterized by a single fluorescence lifetime of 20.5 ns attributed to H1*, the excited-state of the H1 tautomer (Fig. 5A-B). A similar behavior was also observed at pH³11, where the fluorescence decays displayed a single component (13.5 ns) that can be associated to the decay of B*, the excited-state of the B form (Fig. 5D). At pH 10, the decay was nicely fitted by a biexponential function (13.5 and 20.5 ns were fixed), suggesting that both H1* and B* forms emit (Fig 5C ). Within the pH range 6-9, the fluorescence decays were fitted to a two-component model with decay times of 20.5 ns (fixed as obtained from the experiments performed upon excitation at 360 nm) and 12.6 ns (Fig. 6A, B andC). As the DAS of the shorter component corresponded well to the emission spectrum of the H3, this shorter component could be unambiguously attributed to this tautomer. To exclude the possibility of an ultra-fast conversion between the two tautomers in their excited state (i.e. faster than the 40 ps temporal resolution of our TCSPC set-up), TA spectroscopy measurements (80 fs resolution) were performed at pH 7 (lexc=315 nm). The results are shown in Fig. 7A. The TA signal is dominated by a strong excited-state absorption band over the whole spectral range (325-600 nm) which does not show any spectral change, but a monotonous slow decrease with time that is consistent with the long-lived lifetimes of the H1

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Please do not adjust margins and H3 forms. This clearly shows the absence of excited-state conversion between H1 and H3.

At pH 10, the emission and excitation spectra indicate that emission from H1, H3 and B are observed at this pH value. As the lifetime of the H3* (12.6 ns) and B* (13.5 ns) are too close to be resolved, we fixed them (together with the lifetime of H1* i.e. 20.5 ns) in the analysis of the fluorescence decays. From the global analysis of the decays recorded across the emission spectrum of th Gb, we obtained three DAS that match very well with the emission spectra of the three forms (Fig 6D). As for the 360 nm excitation, the fluorescence decay at pH³11 displayed a single component (13.5 ns) that can be attributed to the emission of B* (Fig. 6E).

At pH 2, the measured fluorescence decays were analyzed using the global fit method by sharing the lifetimes (floating parameters) between all wavelengths. Three components were required to accurately fit the measured decays. The two short-lived lifetimes (0.14 and 1.1 ns) exhibited a positive amplitude in the blue part of the spectrum and a negative one in the red part of the spectrum. This suggests the existence of at least one excited-state reaction, where A (that absorbs with a maximum at 309 nm, Fig. 3A) is likely converted into other species, which yield an emission spectrum centered at 481 nm (Fig. 4A). The long-lived component (7 ns) was found to be largely dominant and is likely associated to the species produced through the excited state reaction. To further support this excited-state conversion, TA spectroscopy measurements were performed. In contrast to the data at pH=7, the TA spectrum measured at pH=2 (Fig. 7B, 7C, 7D) shows a clear conversion between two different spectroscopic signatures with a 50-ps timescale (see fig 7D). This conversion is followed by a slow decay, consistent with the 7 ns component measured by TCSPC. A depletion in the early ESA signal is observed around 400 nm (green band at delays < 10ps in Fig. 7B) and is likely associated to the stimulated emission from A*, overlapping the ESA signal in this spectral range. Three components are required to accurately fit the fluorescence decays obtained at pH 3, namely the two short-lived lifetimes (0.14 and 1.1 ns) associated to the excited-state reaction and a long-lived lifetime (14.8 ns). At pH 4, H3 is already present in the ground state (12.6 ns) and is therefore significantly contributing to the fluorescence decays. Hence, the fit of the data provided 4 components (0.14, 1.1, 12.6 and 19.7 ns). At pH 5, the decays were largely dominated by the contributions of H3 (12.6 ns) and H1 (20.5 ns) associated to their direct excitation, so that it was no more possible to detect any short-lived lifetime. Our time-resolved measurements support thus a model in which an excited state reaction is taking place at low pH. As a result, as already reported for other compounds [START_REF] Skilitsi | Ultrafast photophysics of the environment-sensitive 4ʹ-methoxy-3-hydroxyflavone fluorescent dye[END_REF][START_REF] Chou | Femtosecond Dynamics on Excited-State Proton/ Charge-Transfer Reaction in 4'-N , N -Diethylamino-3-hydroxyflavone. The Role of Dipolar Vectors in Constructing a Rational Mechanism[END_REF], the measured fluorescence lifetimes should depend on the lifetimes of the interconverting species and the excited-state kinetic constants that rule their interconversion. This hypothesis would be consistent with the increase of the longest lifetime from 7 ns (pH 2) to 20.5 ns (pH 5). This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins Following fast spectral relaxation within the first 1 ps (likely due to intramolecular and/or solvent relaxation), a time constant of (48±2) ps is determined and assigned to the excited state conversion kinetics of A*.

Calculations

Ground state species and absorption properties

Supporting the results obtained for the th G nucleoside [START_REF] Martinez-Fernandez | Excited-State Dynamics of Thienoguanosine, an Isomorphic Highly Fluorescent Analogue of Guanosine[END_REF], only the H1 and H3 tautomers of th Gb are populated at neutral pH, the former being slightly more stable, in agreement with the experimental observations (Table S1 in SI). The computed vertical absorption energies are in good agreement with the experimental absorption spectra, but with a small overestimation (0.3 eV) of the transition energy of H3 (Table 1).

As a first step to the study of th Gb in basic pH, we have considered the possible species corresponding to the anionic form of th Gb (Fig. S5). As shown in Table S1, the most stable species is predicted to be the keto-amino one, where both N1 and N3 are deprotonated. This form, labelled as [ th Gb] -, should be the only one populated in basic conditions and thus correspond to the B form. This attribution is strengthened by the good match of the computed vertical absorption energy for this species with the experimental absorption maximum at high pH (Table 1).

For acidic solutions, we have characterized the three possible forms for a protonated th Gb (Figure 1a). The most stable (Table S1) is the one where both N3 and N1 atoms are protonated, hereafter labeled as [ th Gb-H1-H3] + . It is at least 0.4 eV more stable than the other two possible forms, suggesting that it should be the only one at low pH and thus correspond to the A form. Moreover, the absorption maximum computed for [ th Gb-H1-H3] + provides an intense transition at 306 nm, in excellent agreement with the experimental absorption maximum at low pH (Table 1). The absorption maxima computed for the two other forms were in lesser agreement with the experiments, supporting our assignment.

Excited state species and emission properties

Geometry optimizations of the lowest energy bright excited state of H1 and H3 (identified as H1* and H3*) lead to stable minima of the Potential Energy Surface (PES), in line with the substantial experimental quantum yield of both tautomers. The computed emission energies and Stokes shift match fairly well with their experimental counterpart (Table 1) [START_REF] Martinez-Fernandez | Excited-State Dynamics of Thienoguanosine, an Isomorphic Highly Fluorescent Analogue of Guanosine[END_REF].

For the emission of th Gb at basic pH, geometry optimization of [ th Gb] - leads to a stable minimum of the PES, characterized by a strong emission at 3.06 eV (f=0.095). We have also optimized the minima of the bright excited state of four other possible forms. Based on their relative stabilities (Table S1), emission energies (Table S2) and associated lifetimes, we can conclude that only the S1 minimum of [ th Gb] -contributes to the fluorescence of th Gb at high pH.

We have next optimized the minima of the spectroscopic state for the three possible species at acidic pH. The computed emission energy for [ th Gb-H1-H3]* + (the most stable in the ground state) is 382 nm. This value matches well with the negative signal appearing at ~400 nm in the early TA spectrum (Figure 7B and 7C), which could be then assigned to stimulated emission from this species. On the other hand, it does not match the steady state emission spectrum, being significantly blue-shifted (Figure 4A). Our calculations predict, in fact, that the bright excited state minimum of the [ th Gb-H1-OH]* + form is more stable by 0.3 eV (Table S1) compared to [ th Gb-H1-H3]* + , suggesting that [ th Gb-H1-OH]* + plays a major role in the emission of th Gb at acidic pH. This conclusion is further substantiated by the computed emission wavelength (479 nm) from the minimum of [ th Gb-H1-OH*] + that almost coincides with the experimental maximum (482 nm) (Table 1). In contrast, the minimum of [ th Gb-H3-OH]* + is the least stable and should not be involved in the photophysics of th Gb.

Finally, by computing the radiative lifetime of the different excited states and using the measured QYs, we predicted the fluorescence lifetimes of the different species (Table 1). Comparison with the experimental lifetimes shows reasonable agreement, correctly predicting that H1 has a longer lifetime than H3. Our calculations are also in good agreement with experiments for the B form; the estimated fluorescence lifetime being close to the experimental value.

Discussion

In this study we explored the photophysical properties of th Gb, a highly emissive G analog, as a function of pH.. At neutral pH, the absorption and emission properties of th Gb were found to be very similar to those previously reported for the th G ribonucleoside [START_REF] Shin | Emissive RNA Alphabet[END_REF] and deoxyribonucleoside [START_REF] Sholokh | Tautomers of a Fluorescent G Surrogate and Their Distinct Photophysics Provide Additional Information Channels[END_REF][START_REF] Park | Highly emissive deoxyguanosine analogue capable of direct visualization of B-Z transition[END_REF][START_REF] Sholokh | Conquering 2-Aminopurine's Deficiencies: Highly Emissive Isomorphic Guanosine Surrogate Faithfully Monitors Guanosine Conformation and Dynamics in DNA[END_REF]. This indicates that as for other fluorescent nucleobases [START_REF] Jones | 2-aminopurine as a fluorescent probe of DNA conformation and the DNA-enzyme interface[END_REF][START_REF] Reichardt | Role of intersystem crossing in the fluorescence quenching of 2aminopurine 2ʹ-deoxyriboside in solution[END_REF], the sugar does not substantially alter the photophysical properties of the aromatic chromophore.

The DAS and deconvoluted emission spectra were used to determine the relative contributions of the different species in the excited state (Fig. 6F). The percentages were corrected using the differences in the extinction coefficients between the different forms (Fig. 3A). At the 6-9 pH range, the percentages of the two tautomers as deduced from the DAS and the deconvoluted emission spectra match well with those obtained from the absorption data, confirming that we have two tautomers in equilibrium in the ground state, with K=[H1]/[H3]=1.3±0.1, and indicating that no interconversion occurs in the excited state. The ground-state equilibrium between H1 and H3 is also fully supported by the ab initio calculations. As each of the two tautomers is excited and emits independently from the other one, the kinetic rate constants governing the equilibrium between the two forms are likely slow with respect to the excited state lifetime. Indeed, no interconversion is observed in the excited state, as shown by the TA data (Fig. 7A) and by the fact that excitation of only the H1 tautomer at wavelengths ³ 350 nm does not generate any emission from the H3 tautomer. A good match is seen for the population of the B, H1 and H3 forms at pH 10, as obtained from the absorption spectra, the deconvoluted emission spectra and the DAS. Moreover, the ratio between the H1 and H3 tautomers is about 1.2 and matches well with the equilibrium constant between the two forms. The data at pH 10 strongly suggest that the equilibrium between the H1 and H3 tautomers and the B form attributed to the [ th Gb] -anion is a ground-state equilibrium. The pKa* value (10.2±0.1) calculated from the DAS (Fig. 6F) is in perfect agreement with the pKa value calculated from the absorption data (10.2±0.1) and computed from the Förster cycle theory or quantum mechanical calculations (Table S4), confirming that the conversion occurs in the ground state. As a result, each of these forms is excited and emits independently, from the two others. Finally, in line with the pKa values, both the absorption and emission spectra indicate that the H1 and H3 tautomers are fully converted into the [thGb] -anion at pH ³ 11.

At acidic pHs, two short-lived lifetimes that displayed positive and negative amplitudes over the studied spectral range were identified, suggesting an excited-state reaction. From quantum mechanical calculations, this reaction was assigned to the conversion of [ th Gb-H1-H3]* + , the excited-state of the [thGb-H1-H3] + into the excitedstate of [ th Gb-H1-OH] + . The computed absorption maximum of [ th Gb-H1-H3] + and emission maxima of both [ th Gb-H1-H3]* + and [ th Gb-H1-OH] +* were consistent with the experimental data. Moreover, [ th Gb-H1-OH] + was found to further convert in the excited state into the H1 tautomer as evidenced by the increase of the long-lived lifetime that reach 20.5 ns at pH 5. Finally, at pH 4 and 5 it was possible to fit the fluorescence decays by including and fixing the lifetime of H3 (12.6 ns), likely indicating that H3 absorbs and emits independently from the other two forms and is therefore not involved in the proton This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins . Excitation and emission wavelengths were at 315 nm and 480 nm, respectively. The solid black lines correspond to the fits using the numerical integration of the differential equation system described in the SI and the values given in Table S6 and scheme 1.

To validate this model, the fluorescence decays in the pH range 2-5 were fitted by numerically solving the set of coupled differential equations given in the SI. The decays used for the fits were recorded with an excitation at 315 nm and an emission at 480 nm, to account for the decays of both [ th Gb-H1-OH]* + and H1* forms. Using our model, it was possible to obtain the time-dependent concentrations of the different species in the excited-state after the initial pulsed excitation (40 ps). By considering that the fluorescence decays measured at 480 nm are proportional to the excited state populations ([ th Gb-H1-OH]* + and H1*) generated by exciting [ th Gb-H1-H3] + plus the fraction of H1 directly excited at 315 nm (relative contribution displayed in Fig. 3B), we were able to fit the fluorescence decays (Figure 8) and determine the kinetic rate constants.

Scheme 2: Proposed photophysics of th Gb in aqueous solution

The kinetic rate constants and fluorescence lifetimes obtained through this approach are gathered in Table S6 and Scheme 1. Interestingly, the calculated lifetime value of H1 is very close to the experimental one. Moreover, the ratio k+/k-was found to change by 10-fold for each pH change by one unit, in full line with the change in proton concentration as predicted by our model. Finally, the model Please do not adjust margins Please do not adjust margins validity was further substantiated by calculating a pKa* value of 1.0 from the ratio of the ESPT rate constants, in reasonable agreement with the calculated pKa* (2.8). Finally, both ESIPT and ESPT reactions occur on sub-50 ps timescale in perfect agreement with values found in literature for organic compounds [START_REF] Skilitsi | Ultrafast photophysics of the environment-sensitive 4ʹ-methoxy-3-hydroxyflavone fluorescent dye[END_REF] , [START_REF] Kumpulainen | Influence of Solvent Relaxation on Ultrafast Excited-State Proton Transfer to Solvent[END_REF].

Conclusions

We have used a combination of spectroscopic techniques and quantum mechanical calculations to investigate the photophysical properties of the th Gb nucleobase at various pHs. This allowed us to identify three new species for th Gb and characterize their photophysics in both the ground-and excited-states (scheme 2). We found that at pH 6-9 the absorption and emission properties were dominated by the previously identified H1 and H3 tautomers, which are in equilibrium in the ground-state and do not interconvert during their fluorescence lifetime. At basic pH, these ground-state tautomers deprotonate into [ th Gb] -(pKa = 10.2). Similarly to neutral pH, no excited-state reaction appears to occur between the different tautomers at basic pH. At acidic pH, the groundstate H1 and H3 tautomers are converted into the protonated and blue-shifted [ th Gb-H1-H3] + species (pKa = 4.3). A more complex picture appears in the excited state at these acidic pHs, as [ th Gb-H1-H3]* + appears to tautomerize to [ th Gb-H1-OH]* + through an ESIPT reaction and is then converted into H1* through a water-mediated ESPT reaction. As the identified species differ by their spectra and their fluorescence lifetime, th Gb may be used as a local probe for sensing pH changes. The probe would be especially useful in the 2-6 pH range, where dramatic photophysical changes can be perceived.

Figure 1 .

 1 Figure 1.a) th Gb forms analyzed in the present study. The formal charge of the acidic and basic resonant forms can be located on atom 2, 9 or 11. For sake of simplicity, the charge is placed in the center of pyrimidine moiety. b) Solvation model considering 6 explicit water molecules.

  . The obtained values (4.3±0.1 and 10.2±0.1) are in excellent agreement with those calculated from the pH dependence of the absorption maxima (4.4±0.1 and 10.3±0.1, Fig. 2C).

Figure 2 :

 2 Figure 2: A) and B) Absorption spectra of th Gb recorded at different pH. C) Dependence of the wavelength of the absorption maximum as a function of pH. The experimental points (red disks) were fitted by the Henderson-Hasselbalch equation (solid line, eq. 1).

Figure 3 :

 3 Figure 3: A) Absorption spectra of the four th Gb species. The spectra of the H1 and H3 forms were from Sholokh et al(14), while the spectra of the acidic and basic forms were deduced from the spectra at pH 2 and 12 in Figure 2. B) Concentration profiles of the different th Gb species in the ground state, as obtained from the deconvolution of the spectra in Figure 2A and B. The solid lines correspond to the fit of the experimental points to equation 1.

Figure 4 :

 4 Figure 4: Emission spectra of th Gb recorded at different pH and excitation wavelengths.

  Fig S2). Indeed, the excitation spectra recorded at 375 nm (Fig S2A) and 450 nm (Fig S2B) overlap well with the absorption spectra of the H3 and B forms, respectively, while the excitation spectrum recorded at 500-550 nm (Fig S2C) is comparable to the absorption spectrum of the H1 tautomer. Similarly to pH 11 and 12, the emission spectra at pH 2 were independent of the excitation wavelength and showed an emission maximum at 482 nm (Fig 4A). In line with the absorption data, the This journal is © The Royal Society of Chemistry 20xx

Figure 5 :

 5 Figure 5: A-D) Decay associated spectra obtained at pH 5, 7, 10 and 12, upon excitation at 360 nm. The spectrally resolved fluorescence decays were analyzed by using the global fit algorithm as described in the main text.

Figure 6 :

 6 Figure 6: A-E) Decay associated spectra determined at pH 6, 7, 9, 10 and 12, upon excitation at 315 nm. The spectrally resolved fluorescence decays were analyzed by using the global fit algorithm as described in the main text. F) Relative concentrations of the different th Gb species in the excited state obtained from the DAS corrected from the differences in the extinction coefficients.

Figure 7 :

 7 Figure7: TA spectroscopy of th Gb upon excitation at λex=315nm: 2D maps of the TA signal (ΔA, color scale) as a function of probe wavelength (nm) and pump-probe delay (ps) measured at A) pH=7 and B) pH=2. The signal is everywhere positive (red), i.e. dominated by strong excited-state absorptions (ESA). The horizontal and vertical dashed lines in panel B respectively illustrate the time delay of 2.5ps at which we plot the transient spectrum in panel C, and the wavelength of 400nm at which we monitor the decay kinetics in panel D. We identify the green region seen in panel B around 400 nm at early time delays to be the stimulated emission (SE, negative signal) of species A*, which is overlapping its positive and more intense ESA at the same wavelength. The result of the global fitting (see "Experimental section") is illustrated in panel D (red line). Following fast spectral relaxation within the first 1 ps (likely due to intramolecular and/or solvent relaxation), a time constant of (48±2) ps is determined and assigned to the excited state conversion kinetics of A*.

  Scheme 1: Photocycle model used to describe the emission properties of th Gb in acidic conditions. The equilibria in the groundstate are not indicated.

Figure 8 :

 8 Figure 8: A-D) Fluorescence decays recorded at pH 2, 3, 4 and 5 (red curves). Excitation and emission wavelengths were at 315 nm and 480 nm, respectively. The solid black lines correspond to the fits using the numerical integration of the differential equation system described in the SI and the values given in TableS6and scheme 1.

th Gb-H1-OH] +

			Theoretical			Experimental
	FC	S1	DE	f	c t	Absorption	Emission	Stokes	c t
		min	(Stokes					Shift
			Shift)
						H1
	3.76	2.99	0.77	0.0668	16	3.73	2.65	1.08	20
	(329)	(414)			(a)	(334)	(470)
						H3
	4.23	3.40	0.83	0.0855	9	3.96	3.10	0.86	13
	(293)	(364)			(a)	(313)	(400)
					[ th Gb-H1-H3] +
	4.05	3.24	0.81	0.0534	-	4.00
	(306)	(382)			(c)	(309)
	[ 3.24 2.58 0.66 0.0460 8		2.57		7
	(382)	(479)			(b)		(482)
					[ th Gb-H3-OH] +
	3.81	3.00	0.81	0.0546	5
	(325)	(412)			(b)
						[ th Gb] -
	3.78	3.06	0.72	0.0949	10	3.64	2.74	0.90	13
	(328)	(405)			(a)	(340)	(451)

Table 1

 1

	. Computed absorption (Franck Condon geometry) and
	emission (S1 min) energies in eV (nm), Stoke shift, oscillator strength
	(f) and radiative lifetimes of different forms of th Gb at the TD-PBE0/6-
	311+G(2d,2p)//PBE0/6-31+G(d,p) level of theory, explicitly including
	6 H2O molecules and PCM into the calculations. Fluorescence
	lifetimes were calculated assuming quantum yield of (a) 0.5 or (b)
	0.1. (c) presumably very low fluorescence QY due to excited state
	reaction, as evidenced by TA and time-resolved fluorescence
	experiments.
	Interestingly, the computed pKa's (3.2 for H1 and 4.2 for H3) are fully
	consistent with the experimental ones, considering the presence of
	two almost equipopulated tautomers at neutral pH and the intrinsic

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 5

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 7

Acknowledgements

We thank Dr Thomas Gustavsson for discussion and preliminary time-resolved femtosecond experiments. This work was supported by the Agence Nationale de la Recherche (ANR blanc PICO2 and SMFLUONA), the Labex NIE and the Centre National pour la Recherche Scientifique (CNRS). Y. M. is grateful to the Institut Universitaire de France (IUF) for support and providing additional time to be dedicated to research. YT thanks the National Institutes of Health for support (via grant number GM 069773).

Conflicts of interest

There are no conflicts to declare.


Dima Grigoriev
email: dmitry.grigoryev@univ-lille.fr

Danylo Radchenko
email: danradchenko@gmail.com

ON A TROPICAL VERSION OF THE JACOBIAN CONJECTURE

We prove that, for a tropical rational map if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative). In addition, for a tropical rational map we prove that if the Jacobians have the same sign and if its preimage is a singleton at least at one regular point then the map is an isomorphism.

Introduction

We provide two criteria for a tropical map of n-dimensional real spaces to be an isomorphism. The basic concepts of tropical (or min-plus) mathematics can be found in [START_REF] Maclagan | Introduction to Tropical Geometry[END_REF]. The first criterion concerns tropical rational maps and states that if at any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism (Proposition 1 and Theorem 2). Note that there is in total a finite number of Jacobian matrices corresponding to polyhedra on which the tropical rational map is linear. We also show that this criterion is not necessary (even for 2-dimensional tropical polynomial isomorphisms).

The second criterion deals with tropical polynomial maps and holds for 2dimensional real spaces. It claims that if the Jacobians at all the smooth points have the same sign (either positive or negative) then the tropical polynomial map is an isomorphism (Theorem 5). We construct an example which demonstrates that this criterion fails for tropical rational 2-dimensional maps as well as for tropical polynomial 3-dimensional ones. In addition, for a tropical rational map we prove that if the Jacobians at all the smooth points have the same sign and if its preimage is a singleton at least at one regular point then the map is an isomorphism (Theorem 4).

Weak version of the tropical Jacobian conjecture

By a tropical algebraic rational function (in n variables) we mean a function from R n to R of the form min{A 1 , . . . , A k } -min{B 1 , . . . , B l } where A 1 , . . . , A k , B 1 , . . . , B l are linear functions with rational coefficients (when the coefficients at the variables are positive integers one talks about tropical rational functions). They play the role of a tropical analog of algebraic functions (cf. [START_REF] Grigoriev | Tropical Newton-Puiseux polynomials[END_REF]). In fact, one could stick with real coefficients since no proofs use their rationality.

From a different point of view, defining a tropical algebraic rational function amounts to giving a finite decomposition R n = i C i , where each C i is a closed n-dimensional polyhedron and the interiors of C i and C j do not intersect if i = j, and specifying a linear function f i on each C i in such a way that f i and f j agree on C i ∩ C j . A similar description applies in general to tropical algebraic rational mappings from R n to R m . We will call (f i , C i ) or just f i the linear pieces of f .

Observe that if a tropical algebraic rational map R n → R n is a bijection then its inverse is also a tropical algebraic rational map ( [START_REF] Kripfganz | Piecewise affine functions as a difference of two convex functions[END_REF], [START_REF] Ovchinnikov | Max-min representation of piecewise linear functions[END_REF], cf. also [START_REF] Grigoriev | Tropical Newton-Puiseux polynomials[END_REF] where one can find further references). In this case we call such a map an isomorphism.

Proposition 1. If a tropical algebraic rational map f : R n → R n is locally injective, then it is an isomorphism.

Proof. Let f 1 , . . . , f N be the linear pieces of f . Since f is locally injective, each f i has to be invertible, and hence for any compact

K ⊂ R n we have that f -1 i (K) is compact. Therefore, f -1 (K) ⊆ N i=1 f -1
i (K) is compact, and thus f is a proper mapping. By the domain invariance theorem [START_REF] Brouwer | Zur Invarianz des n-dimensionalen Gebiets[END_REF] the mapping f is locally a homeomorphism. The result then follows directly from [5, Th. 2] that, specialized to our setting, says that if a continuous mapping f : R n → R n is a local homeomorphism and is proper, then it is a global homeomorphism. This result can be treated as a weak tropical version of the Jacobian conjecture in the sense that for tropical rational algebraic maps local invertibility implies global invertibility. The problem remains how to tell if a given map is locally an isomorphism. There is a simple sufficient condition for local invertibility of Lipschitz mappings that can be easily stated in our case (it is not hard to show that tropical rational algebraic maps are Lipschitz). Let f : R n → R n be a tropical algebraic rational mapping with linear pieces

(f i , C i ), i = 1 . . . , N . For any point x ∈ R n let I = {i | x ∈ C i }.
We then define ∂ f (x) to be the convex hull of the set of differentials (Jacobian matrices) of f i , i ∈ I (in the space of matrices Mat n×n (R)). The following is a special case of the inverse function theorem for Lipschitz mappings proved by Clarke [2, Th. 1].

Theorem 2. If f : R n → R n is a tropical algebraic rational mapping and the set ∂ f (x 0 ) contains only nonsingular (i. e. with non-vanishing determinants) matrices, then f is a homeomorphism in a neighborhood of x 0 .

Note, however, that the sufficient condition in Theorem 2 for f to be an isomorphism is not necessary, as the following example shows.

Example 3. We construct a tropical polynomial isomorphism f : R 2 → R 2 as a composition of two tropical polynomial maps (isomorphisms): a lower-triangular map (x, y) → (x, y + min{αx, βx}), α < β and an upper-triangular one:

(x, y) → (x + min{ay, by}, y), a < b.

Then f (x, y) = f 1 := (x + a(y + αx), y + αx) if x > 0, y + αx > 0; f 2 := (x + b(y + αx), y + αx) if x > 0, y + αx < 0; f 3 := (x + a(y + βx), y + βx) if x < 0, y + βx > 0; f 4 := (x + b(y + βx), y + βx) if x < 0, y + βx < 0.
The corresponding matrices of differentials at the origin are as follows:

1 + aα a α 1 , 1 + bα b α 1 , 1 + aβ a β 1 , 1 + bβ b β 1 .
The sum of the second and the third matrices is singular when (β -α)(b-a) = 4 (in particular, one can put β = b = 2, α = a = 0). Thus, ∂ f (0, 0) contains a singular matrix.

Strong version of the tropical Jacobian conjecture

Denote by J i , 1 ≤ i ≤ N the determinants (Jacobians) of the differentials of f i . If f is an isomorphism then all J i must have the same sign (either positive or negative). This follows from a general result about homeomorphisms (see, for example, [4, Th. 5.22]) or can be verified directly by comparing the Jacobians J i , J j for a pair of f i , f j defined on adjacent (n-dimensional) polyhedra separated by a common (n -1)-dimensional facet. Observe that the condition on ∂ f (p) in Theorem 2 implies that all the Jacobians have the same sign since for a pair of real matrices A, B with det(AB) < 0 there exists 0 < t < 1 such that det(tA

+ (1 - t)B) = 0.
The condition that all J i are nonzero and have the same sign is not sufficient to guarantee that f is an isomorphism (see Example 6 below), but we will now show that it is almost sufficient. In analogy to the case of smooth maps, we call y ∈ R n a regular value of f , if it does not belong to the set N i=1 f (∂C i ). A version of Sard's theorem is true (and is trivial) in this case: the set of regular values is dense in R n . Theorem 4. A necessary and sufficient condition for a tropical algebraic rational mapping f : R n → R n to be an isomorphism is that all Jacobians J i have the same sign, and that |f -1 (y 0 )| = 1 for at least one regular value y 0 .

Proof. The necessity is clear. To prove sufficiency we will use standard techniques from topological degree theory (for a general reference, see [8, Ch. IV]).

Without loss of generality we can assume that all Jacobians are positive. Since f is proper (see the proof of Proposition 1), it has a well-defined degree deg(f ) (as a mapping from the one-point compactification of R n to itself), and by computing it using the preimage of the regular value y 0 we get deg(f ) = 1. This, in turn, together with the positivity of all Jacobians implies that |f -1 (y)| = 1 holds for any regular value y. The proof would then be complete if we could show that f is an open mapping: indeed, if f (x 1 ) = f (x 2 ) for some x 1 = x 2 , then |f -1 (y)| ≥ 2 for all y in a sufficiently small neighborhood of f (x 1 ), which would contradict the above computation for regular y.

To show that f is open we use an argument similar to the one used in [10, Proof of Thm. 3]. Note that |f -1 (y)| ≤ N for all y, where N is the number of linear pieces of f . Therefore, for any x ∈ R n we can find a small neighborhood U of x such that f -1 (f (x)) ∩ U = {x}. Let V be a small open ball around f (x) that does not intersect f (∂U ). Then for some smooth point x 0 ∈ U ∩ f -1 (V ) we have that f (x 0 ) is a regular value, and hence deg(f, U, f (x 0 )) = 1. Since deg(f, U, p) only depends on the connected component of R n f (∂U ) to which p belongs, we get that deg(f, U, y) = 1 for all y ∈ V . But if deg(f, U, y) = 0, then y ∈ f (U ) (see [8, Cor. IV.2.5(3)]), so that V ⊂ f (U ). Thus f is an open mapping.

Let us remark that this result gives a simple computational approach to checking that a given tropical rational algebraic map f is an isomorphism. Indeed, to check the above conditions one only needs to compute the set of linear functions f i and the preimage of a generic point y 0 , but both of these tasks are easy to do using linear programming.

It turns out that if n = 2 and f is a tropical algebraic mapping, so that each coordinate of f is concave and piecewise linear, then a stronger result is true.

Theorem 5. If f : R 2 → R 2 is a tropical algebraic mapping (i.e., f = (φ 1 , φ 2 )
where φ i are tropical algebraic functions) such that all the Jacobians J i have the same sign, then f is an isomorphism.

Proof. For a ∈ R such that the preimage φ -1 1 (a) ⊂ R 2 is non-empty, it is the boundary of the convex polygon P := {x | φ 1 (x) ≥ a}. Consider the supporting to P lines L 1 , . . . , L m . Either m = 2 and L 1 , L 2 are parallel or one can renumber L 1 , . . . , L m to make their slopes in the clock-wise order (with respect to an arbitrary point of P ).

In the latter case φ -1 1 (a) is a connected polygonal line with the consecutive edges (some of them, perhaps, unbounded) lying on L 1 , . . . , L m , respectively. Due to the condition imposed on the Jacobians J i in Theorem 5, φ 2 is strictly monotone along φ -1 1 (a). Therefore, φ -1 1 (a) has to be unbounded with just two unbounded edges L 1 , L m (or, in case m = 1 one unbounded edge L 1 being the boundary of a halfplane P ), and φ 2 restricted to φ -1 1 (a) (being a piecewise linear function with a finite number of pieces) provides an isomorphism with R. Thus, we conclude that f is an isomorphism if φ -1 1 (a) is connected (or empty) for all a ∈ R by Proposition 1. By symmetry, f is also an isomorphism if φ -1 2 (a) is connected (or empty) for all a ∈ R. Now we study the case when m = 2 and L 1 , L 2 are parallel. Let φ i =: min{A i,1 , . . . , A i,si }, 1 ≤ i ≤ 2. Then each linear function A 1,q , 1 ≤ q ≤ s 1 is constant on L 1 . Hence the graph Φ 1 ⊂ R 3 of φ 1 is the Cartesian product of L 1 and a convex polygonal line Q ⊂ R 2 which can be treated as the graph of a univariate concave piecewise linear function. The latter function is bounded from above since Φ 1 contains L 1 , L 2 . Therefore, φ 1 attains its maximum on a line L parallel to L 1 , L 2 .

Thus, it remains to consider the case when for both φ 1 , φ 2 there exist a 1 , a 2 ∈ R such that φ -1 1 (a 1 ), φ -1 2 (a 2 ) are disconnected, so they consist of pairs of parallel lines. So, L separates two adjacent pieces on which φ 1 is linear with their differentials being collinear with a negative coefficient of collinearity. If at some point y of L the piecewise linear function φ 2 is non-singular (thereby, is linear in a neighborhood of y) then (the only) two linear maps f i , f j defining f in a neighborhood of y have the Jacobians J i , J j with opposite signs contrary to the condition of Theorem 5.

Otherwise, if φ 2 has an edge on L separating two pieces on which φ 2 is linear, then for two linear maps f i , f j which define f in a neighborhood of any point of this edge their Jacobians J i , J j both vanish. Obviously, φ 1 can't attain its maximum on a 2-dimensional facet since the differential of φ 1 would vanish on it. We get a contradiction with the supposition that there exist a 1 , a 2 ∈ R such that φ -1 1 (a 1 ), φ -1 2 (a 2 ) are disconnected. The simple condition of Theorem 5 is no longer sufficient for n ≥ 3, as the following example shows. Example 6. We will construct a tropical polynomial mapping f : R n → R n , n ≥ 3, for which the Jacobians J i > 0 for all i, but which is not an isomorphism.

Clearly, it is enough to construct such an example for n = 3. We start with a two-dimensional tropical rational mapping g : R 2 → R 2 defined by (x, y) → (|x| -|y|, |x + y| -|x -y|).

Note that the Jacobian of each linear piece of g is equal to 2, but g(x, y) = g(-x, -y), so that g is not injective. To construct f we start with the mapping that sends (x, y, z) to (g(x, y), z) and then apply elementary piecewise linear transformations to get rid of all the subtractions in the min-plus form. More precisely, let us define h : R One can check that h is a tropical Laurent polynomial map (since -|x| = min(-x, x) is), its Jacobian is equal to 2 at every smooth point, and h(x, y, z) = h(-x, -y, z). To get the mapping f , we compose h with z → z + 2x + 2y to obtain f (x, y, z) = (min(2y, 0) + min(2x + 2y, 0) + z + x, min(2x, 0) + min(2x, 2y) + z + y, min(2x, 0) + min(2x + 2y, 0) + z + y).

Then f is a tropical polynomial map whose Jacobian at every smooth point is equal to 2, but it is not an isomorphism since f (-x, -y, z + 4x + 4y) = f (x, y, z).

3 →

 3 R 3 by (x, y, z) → (-|y| -|x + y| + z, -|x| -|x -y| + z, -|x + y| -|x| + z).

Acknowledgements. The first author is grateful to the grant RSF 16-11-10075 and to MCCME for inspiring atmosphere. The second author is grateful to Max Planck Institute for Mathematics in Bonn for its hospitality and financial support.