Andrei S Kozlov

Thomas Risler

Armin J Hinterwirth

A J Hudspeth

Relative stereociliary motion in a hair bundle opposes amplification at distortion frequencies

Keywords: DIC, differential interference contrast, HEPES, 4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid, RMS, root mean square

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Relative stereociliary motion in a hair bundle opposes amplification at distortion frequencies

INTRODUCTION

The sensitivity and frequency selectivity of hearing are controlled by the physics of mechanoelectrical transduction in the ear's receptor, the hair cell. Each hair cell bears on its apical surface a bundle of cylindrical stereocilia separated by small gaps filled with viscous endolymph. Stereocilia are connected to their closest neighbors by several protein linkages.

Among these, the tip links connect the tip of each stereocilium to the side of its tallest neighbor and translate hair-bundle deflection into tension that controls the open probability of mechanosensitive transduction channels [START_REF] Hudspeth | How the ear's works work[END_REF]. Coherent motion of the stereocilia in a hair bundle minimizes viscous energy dissipation and preserves at the cellular level the mechanical nonlinearity that originates from individual channel molecules in the form of gating compliance [START_REF] Howard | Compliance of the hair bundle associated with gating of mechanoelectrical transduction channels in the bullfrog's saccular hair cell[END_REF][START_REF] Kozlov | Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale[END_REF]. Coupled to a source of energy through myosin motors, gating compliance contributes to an active process that lowers the threshold of sound detection by counteracting the dissipative influence of the surrounding liquid (reviewed in [START_REF] Hudspeth | How hearing happens[END_REF][START_REF] Fettiplace | Clues to the cochlear amplifier from the turtle ear[END_REF][START_REF] Lemasurier | Hair-cell mechanotransduction and cochlear amplification[END_REF][START_REF] Fettiplace | The sensory and motor roles of auditory hair cells[END_REF][START_REF] Hudspeth | Making an effort to listen: mechanical amplification in the ear[END_REF]. Together, these features make the hair bundle a highly sensitive mechanical detector and amplifier.

A negative consequence of direct channel gating is that the ear distorts and sometimes suppresses external sounds. In particular, the ear can produce new vibrations at frequencies that correspond to integer combinations of those present in the stimulus. Such internally generated distortion products interact with external signals at other locations along the cochlea, creating additional distortion products that interact further with the input. The result is a cascade of oscillations, all internally generated by the ear and absent from the original sound [START_REF] Robles | Two-tone distortion on the basilar membrane of the chinchilla cochlea[END_REF][START_REF] Jülicher | Physical basis of two-tone interference in hearing[END_REF]. The cause of these distortions is the nonlinear response that originates in the direct gating of the transduction channels by mechanical force.

Unidirectional amplification, in which the active movements of hair bundles do not feed back onto the basilar membrane, can in principle decrease distortion in a mammalian ear poised at a Hopf bifurcation [START_REF] Reichenbach | Unidirectional mechanical amplification as a design principle for an active microphone[END_REF]. Using a protocol that mimics in vivo stimulation, we have investigated the production of distortion products to ascertain whether a hair bundle can amplify stimuli while precluding amplification of the distortions.

METHODS

Experimental preparation

Using a protocol approved by the Institutional Animal Care and Use Committee of The Rockefeller University, we performed experiments on sacculi from adult bullfrogs (Rana catesbeiana) of both sexes. After having been immersed in water at 0 ˚C, each animal was doubly pitched and decapitated, a procedure considered acceptable by the Panel on Euthanasia of the American Veterinary Medical Association (www.avma.org/issues/animal_welfare/euthanasia.pdf.). The sacculi were then dissected as described previously (Kozlov et al., 2007) and maintained in oxygenated saline solution comprising 120 mM NaCl, 2 mM KCl, 1 mM CaCl 2 , 10 mM D-glucose, and 5 mM 4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid (HEPES) at pH 7.3.

Following a 30-60 min digestion at room temperature with 1 mg•ml -1 collagenase (type XI, Sigma Chemical Co.), the otolithic membrane was removed and the sensory epithelium of the sacculus was peeled from the underlying connective tissue. The epithelium was then folded and secured in an experimental chamber by a golden electron-microscopic grid. All experiments were performed at a room temperature of 20 ˚C to 25 ˚C in the saline solutions used during dissection.

Stimulation

We stimulated each hair bundle with a glass pipette about 1 µm in internal tip diameter and attached only to the kinociliary bulb. A larger pipette attached to the tall edge of a hair bundle entirely suppressed the distortion products there [START_REF] Kozlov | Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale[END_REF], but that was undesirable in this study for two reasons. First, in vivo only the kinociliary bulb is attached to the otolithic or tectorial membrane and displaced, so our experimental arrangement resembled the physiological configuration. Second, to characterize the phase angles at the distortion frequencies we required that these distortions be present at both edges of a hair bundle.

Dual-beam differential interferometer

A dual-beam interferometer allowed the detection of picometer-scale movements by transparent microscopic structures. Based on an earlier single-beam design [START_REF] Denk | Mechanical properties of sensory hair bundles are reflected in their Brownian motion measured with a laser differential interferometer[END_REF][START_REF] Denk | Optical measurement of picometer displacements of transparent microscopic objects[END_REF], it featured two independently controlled beams, each of which acted as an interferometer (Fig. 1). A conventional upright microscope with differential-interference-contrast (DIC) capability (BX51WI, Olympus) served as the basis of the system. The two interferometers were kept independent by assigning a different wavelength to each: a 633 nm, 1.8 mW red He-Ne gas laser (117A, Spectra-Physics) provided one interferometric beam and a 532 nm, 10 mW green diode-pumped solid-state laser (85 GCA 010, Melles Griot) the other.

For each laser, an optical isolator (IO-3D-λ-PHE, OFR Thorlabs Inc.) prevented intensity oscillations resulting from back-reflections into the laser cavity. The light intensity could be modified as necessary with a compensated attenuator (925B, Newport). The beam profile was spatially filtered by coupling it to a single-mode optical fiber. At the fiber's exit port, a singlemode fiber coupler (F-91-C1, Newport) provided a parallel beam of a diameter matching the aperture of the microscope's condenser lens. Passing the beam through a polarizer and a quarterwave plate rendered it circularly polarized before it entered the microscope.

The beams shared the second lens of a beam-steering telescope. The first components of this steering assembly were positioned such that second lens imaged them in the condenser's aperture plane. This arrangement rendered a translation of the steering lenses into a translation of the lasers' focal points in the specimen plane. Each steering lens could be moved either manually with a micrometer-controlled linear stage for coarse positioning of the focal spot or with a controllable piezoelectrical actuator (P-263, PiezoJena) for fine positioning and calibration.

The two parallel beams were directed into the microscope by a dichroic mirror situated below the condenser. At this position infrared light from a conventional halogen lamp was combined with the laser beams to illuminate the specimen. To facilitate positioning of the beams on a hair cell, the infrared image was acquired by a charge-coupled-device camera and displayed on a video monitor.

We used two identical 40x water-immersion objectives of numerical aperture 0.8 (LUMPlanFI/IR, Olympus) in a symmetrical configuration to provide both a condenser that focused the beams to a diffraction-limited spot in the specimen plane and an observation lens.

The condenser stage of the microscope was modified to accommodate the objective lens as well as a DIC prism (WI-DICT2, Olympus) located in its aperture plane. The lower DIC prism split each circularly polarized beam into two sub-beams that were polarized orthogonally to each other and spatially separated by about 100 nm at the specimen. The sub-beams were then recombined in a second, identical DIC prism situated above the objective lens. If both sub-beams encountered optical paths of the same length on their passage through the specimen, their recombination yielded a circularly polarized beam. If one of the sub-beams instead experienced an optical path of a different length as a result of variations in refractive index or specimen thickness, an elliptically polarized beam ensued after recombination.

To measure the ellipticities of the red and green laser beams, each was isolated with a dichroic mirror and passed through a narrow bandpass filter transmissive for the appropriate wavelength. This arrangement ensured that there was no measurable cross-talk between the two channels. Each polarized beam was then split into two orthogonal components by a polarizing beam splitter. Photodiodes attached to transimpedance amplifiers with a roll-off frequency of about 20 kHz reported the intensity of each E-vector component. The analog signals were filtered with eight-pole Butterworth filters with a low-pass corner frequency of 20 kHz, then digitized with 16-bit resolution at sampling intervals of 10 µs. A computer calculated the final deflection values from the detector output for each channel by taking the ratio of the difference between the two E-vector components to their sum. This ratio was directly related to the phase difference between the two sub-beams, which in turn depended on the position of the object that the two sub-beams traversed.

During the initial calibration, pixel values of the video system were converted into distances by use of a stage micrometer with 10-µm increments. A further calibration step related the voltage applied to the piezoelectric actuator that moved either beam's steering lens with the ensuing movement of the respective focal spot. To measure the distance of movement, the focal spot was imaged with the video system and its centroid determined (Image Processing Toolbox, MATLAB). This centroid was tracked during the calibration procedure and used as the position coordinate of the focal spot.

Data acquisition

A single measurement consisted of a set of twenty records, each lasting either 0.1 s or 1 s. Each record corresponded to a double time series, acquired simultaneously and independently with the two laser beams directed at desired positions on a hair cell. Occasionally, when small debris in the optical path or laser pointing instabilities corrupted the signal, a record was rejected on the basis of an unstable root-mean-square amplitude, an abnormal cross-correlation amplitude, or a pronounced drift in the raw time series.

Calibration was achieved during an experiment by translating the focal spot of each laser beam across the specimen by a known distance while recording the resulting detector signals.

Because the calibration factor between focal-spot movement and detector output depended on exactly where the beam was positioned, the detection apparatus was calibrated independently for each beam before and after each record.

Spectral analysis

In order to characterize the degree of common and relative stereociliary motion within the hair bundle, we measured simultaneously the time-dependent positions of stereocilia at two different locations in a hair bundle. From these measurements, we derived the spectral coherency function, a normalized, complex-valued quantity. The modulus, called the coherence, is a real number between zero and one that characterizes the degree to which two objects move together at each frequency. The phase characterizes the delay between the two time series at each frequency. For each of these quantities, spectral estimates were derived by the Thomson multitaper method with six orthogonal tapers and confidence intervals were obtained using jackknife estimation methods (Thomson, 1982;Thomson and Chave, 1991;Percival and Walden, 1993). The justification for this method and further details can be found in the Supplementary Information.

A central assumption underlying spectral-estimation methods is that the time series under consideration are stationary. Nonstationarities in the data were controlled by three means. First, we acquired relatively short time series between which the measurement apparatus was recalibrated to compensate for slow drift. This approach provided a set of independent records that could be used to estimate the desired quantities by an approach similar to overlappedsegment averaging. Next, our criteria for outlier rejection ensured that strongly nonstationary records were eliminated. Finally, the multitaper spectral-estimation technique prevented lowfrequency artifacts from contaminating the estimated spectral densities.

Statistical estimates over different records and measurements

For each measurement in this study, we selected a variable number

€ f 0 -W , f 0 + W [ ] (Supplementary Information).
For statistical purposes, the resulting € P = K × B spectral estimates were considered as independent and equivalent estimates of the same stochastic process. The jackknife variance estimate was then performed on this entire set of realizations.

For each experimental condition, several measurements apiece were performed on multiple cells. The spectral estimate associated with the set of results corresponding to each experimental condition is presented as the mean ± standard deviation for each frequency.

RESULTS

Using the whole-epithelium preparation from the bullfrog's sacculus (Kozlov et al., 2007), we examined the coherency of stereociliary motion in mechanically excited hair bundles using the dual-beam differential interferometer (Fig. 1). To stimulate each hair bundle, we used a glass pipette attached by suction to the kinociliary bulb. The stiff glass probe was displaced with a piezoelectrical device simultaneously at 90 Hz and 115 Hz, a procedure that evokes distortion products at well-defined frequencies [START_REF] Jaramillo | Auditory illusions and the single hair cell[END_REF][START_REF] Kozlov | Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale[END_REF]. We used displacements comparable to those that occur in vivo in response to strong physiological stimuli, typically 50-100 nm in root-mean-square (RMS) magnitude.

We examined the magnitude and phase of the relative motions between stereocilia adjacent to the stimulus probe and those on the opposite side of the hair bundles both at the frequencies of stimulation and at those of the internally generated distortion products. Six stimulated hair bundles displayed a coherence of essentially unity and a phase of approximately zero at the frequencies of stimulation: the coherence and phase at 90 Hz were 0.9992 ± 0.0005 and -0.023 ± 0.034 rad, and those at 115 Hz were 0.9992 ± 0.0005 and -0.027 ± 0.047 rad (Fig. 2). In contrast, at the distortion-product frequencies the coherence dropped sharply and the motion developed a pronounced phase lag. For example, at the distortion frequency f 2 + f 1 = 205 Hz the coherence decreased to 0.51 ± 0.19 and the phase lag was as large as -1.65 ± 2.08 rad.

In principle, differences in signal amplitude can artificially affect the coherence by changing the signal-to-noise ratio when noise unrelated to hair-bundle motions is present. A comparison of the coherence and spectral power at the distortion frequencies with those at immediately adjacent frequencies corresponding to the background noise shows that the reduced coherence of the distortion components is associated with an increased spectral power (Fig. 2).

Because a poor signal-to-noise ratio would produce the opposite correlation, this result indicates that the drop of coherence and the large phase lags at the distortion frequencies are not noise artifacts but are characteristics of the hair-bundle kinematics. In addition, a poor signal-to-noise should yield phases that are ill-defined but of zero mean. That we observe instead large mean phases at the distortion frequencies buttresses the previous affirmation. Such results in the coherence and phase spectra indicate the presence of highly dissipative modes of motion at the distortion components [START_REF] Kozlov | Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale[END_REF]. Note that the phase lags at the various distortion frequencies (Table 1) were correlated with the amount of force that the hair bundle exerted against the stimulating probe at those frequencies [START_REF] Jaramillo | Auditory illusions and the single hair cell[END_REF].

Forces in the tip links, including those originating from channel gating, can change the separation between adjacent stereocilia. Although these relative movements are tiny, they can have a dramatic effect on energy dissipation by exciting modes of motion with high viscous drag [START_REF] Kozlov | Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale[END_REF]. The decrease in coherence observed at the distortion frequencies suggested that it might also be possible to detect a drop caused by the tip links in unstimulated hair bundles undergoing free fluctuations. Because tip links connect stereocilia only along the axis of mechanosensitivity, we compared the coherence for a hair bundle measured along this axis with that obtained in the orthogonal direction. By performing experiments on hair cells confirmed to have functional mechanotransduction by the presence of spontaneous oscillations, we selected hair bundles with intact tip links.

In control experiments on 17 hair bundles, the coherence between 100 Hz and 5 kHz for the same-stereocilium measurements was 0.97 ± 0.04. When measured at the opposite sides of 29 hair bundles under identical conditions, the coherence fell to 0.92 ± 0.05 (Fig. 3). For movements in the direction orthogonal to the axis of mechanosensitivity, the average coherence in this frequency range for the opposite sides of seven hair bundles was 0.97 ± 0.02 (Fig. 3) versus 0.98 ± 0.01 for same-stereocilium measurements in the identical cells. In every instance the mean phase value was close to zero at all frequencies.

The comparison shows that the coherence is higher for motions in the orthogonal plane than for those in the plane of mechanosensitivity (P = 0.0048 by a two-tailed t-test). Therefore, even though tip links elastically connect adjacent stereocilia, they globally disorder stereociliary motion in a hair bundle on a sub-nanometer scale. This seemingly paradoxical observation accords with quantitative predictions of a detailed finite-element model of the bullfrog's saccular hair bundle [START_REF] Kozlov | Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale[END_REF].

The magnitude of thermal movement in the direction orthogonal to the axis of mechanosensitivity for these seven cells, 9.6 ± 1.5 nm RMS, exceeded that in the direction of mechanosensitivity in the 29 cells confirmed to have intact mechanotransduction, 7.3 ± 3.2 nm (P = 0.08). To test whether these differences in signal amplitude could have artificially affected the coherence, we investigated whether there was any significant correlation between the coherence and magnitude values. Correlation coefficients were insignificant for both the cells observed along the axis of mechanosensitivity (correlation coefficient -0.002, P = 0.99) and those observed in the orthogonal direction (correlation coefficient -0.1, P = 0.83). This further validates the comparison of the coherence between the two orientation axes of a hair bundle.

DISCUSSION

A lack of sufficiently accurate experiments has fostered confusion about the degree of relative stereociliary motion and the roles of interstereociliary linkages and the surrounding liquid in coupling stereocilia. Theoretical analyses imply that the thermal fluctuations of stereocilia with differing lengths display distinct spectral characteristics [START_REF] Svrcek-Seiler | Micromechanical models for the Brownian motion of hair cell stereocilia[END_REF]. In accordance with this idea, modeling studies suggest that the tension in tip links is not evenly distributed across a hair bundle, but instead decreases along a stereociliary file [START_REF] Duncan | A finite-element model of inner ear hair bundle micromechanics[END_REF][START_REF] Cotton | Computational models of hair cell bundle mechanics: II. Simplified bundle models[END_REF]Nam et al., 2007aNam et al., , 2007b)). These results notwithstanding, morphometric analysis indicates that the stereocilia are tightly packed against one another at rest, without separation along the axis of mechanosensitivity [START_REF] Jacobs | Ultrastructural correlates of mechanoelectrical transduction in hair cells of the bullfrog's internal ear[END_REF].

Microscopic observations and video measurements disclose that stereocilia do not separate significantly when hair bundles are subjected to large, low-frequency stimuli [START_REF] Hudspeth | Sensitivity, polarity, and conductance change in the response of vertebrate hair cells to controlled mechanical stimuli[END_REF][START_REF] Karavitaki | Sliding adhesion confers coherent motion to hair cell stereocilia and parallel gating to transduction channels[END_REF] and even to rapid displacement steps [START_REF] Crawford | Activation and adaptation of transducer currents in turtle hair cells[END_REF]; but see [START_REF] Duncan | Distal separation of chick cochlear hair cell stereocilia: analysis of contact-constraint models[END_REF]. Furthermore, the stiffness of a hair bundle is proportional to the number of its stereocilia, which suggests an equal distribution of force among them [START_REF] Howard | Stiffness of sensory hair bundles in the sacculus of the frog[END_REF]. Although interstereociliary connections are commonly thought to transmit mechanical forces across a hair bundle and to allow it to move coherently [START_REF] Goodyear | The ankle-link antigen: an epitope sensitive to calcium chelation associated with the hair-cell surface and the calycal processes of photoreceptors[END_REF]Richardson, 1999, Nam et al., 2006), fluid-dynamical calculations emphasize the importance of the viscous interactions between stereocilia and the liquid [START_REF] Zetes | Fluid-structure interaction of the stereocilia bundle in relation to mechanotransduction[END_REF]. The disagreements among these studies stem from the difficulty of resolving very small amplitudes of stereociliary movement and from the uncertainties intrinsic to modeling mechanically complex, many-body systems such as the hair bundle.

Measurements obtained with a dual-beam interferometer indicate that the relative motions in hair bundles are tiny compared to the common motion over a broad range of frequencies. This result obtains both when the hair bundles fluctuate freely (Kozlov et al., 2007) and at the stimulus frequencies when they are excited mechanically [START_REF] Kozlov | Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale[END_REF].

Although morphological differences between the hair bundles of mammals and those of other vertebrates could lead to quantitatively different results, modeling of the stereociliary motion in mammalian bundles suggests a similar behavior [START_REF] Kozlov | Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale[END_REF]. Considering the advantages conferred to the ear by the coherent motion of stereocilia, it would not be surprising if such a mode of operation is a general feature of all hair bundles. Indeed, we found that auditory hair bundles in the tokay gecko also show highly concerted motion of their stereocilia (Supplementary Information).

We have demonstrated that hair bundles display a different behavior at the frequencies of the internally generated distortion products, for which squeezing modes of motion between adjacent stereocilia become significant. This can be seen from the presence at these frequencies of strongly non-zero mean phases indicative of the anti-phase components in the responses. The principal cause of this kinematic behavior is geometrical: because tip links are oriented obliquely to the long axes of the stereocilia, they pull adjacent stereocilia toward one another when tension increases and allow them to separate when it decreases. Even in unstimulated cells this decreases the coherence in the direction of mechanosensitivity as compared with that in the orthogonal direction. Because the relative motions occur at sub-nanometer amplitudes, they are not readily detectable by diffraction-limited imaging techniques [START_REF] Karavitaki | Sliding adhesion confers coherent motion to hair cell stereocilia and parallel gating to transduction channels[END_REF].

The amplitudes of the relative modes of stereociliary motion decrease with frequency owing to the viscous coupling between stereocilia [START_REF] Kozlov | Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale[END_REF]. At sufficiently high frequencies, typically in the kilohertz range, the tip links should not be able to cause relative motions and to lower the slope of the displacement-open probability relation at the distortion frequencies. Consistent with this idea, a cascade of distortion products occurs at the cochlea's high-frequency base [START_REF] Robles | Two-tone distortion on the basilar membrane of the chinchilla cochlea[END_REF]. At the cochlear apex, however, where the nonlinearities are believed to be milder [START_REF] Robles | Mechanics of the mammalian cochlea[END_REF]), the internal damping described here could diminish the amount of distortion.

In a stimulated hair bundle the common mode of motion completely dominates any relative mode at the stimulus frequencies. In contrast, at the distortion frequencies, internally generated squeezing modes are of major importance. Although the active process can counteract the rather modest viscous drag for the common mode at the stimulus frequencies, at the distortion frequencies the hair bundle breaks into a serial arrangement of stereocilia with pronounced squeezing modes that have a thousand-fold higher drag compared to that of the common mode [START_REF] Kozlov | Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale[END_REF]. Amplification can operate at the frequencies of stimulation at which the bundle moves coherently, but should be impaired at the distortion frequencies by independent motions of the stereocilia that decrease the slope of the relation between displacement and open probability and thus limit the sensitivity of mechanoelectrical transduction. It follows that clustering stereocilia in a hair bundle provides a double advantage to the ear. First, it suppresses drag at the stimulus frequencies and thus facilitates amplification by the active process [START_REF] Kozlov | Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale[END_REF]. And second, it increases the drag and decreases the coherence at the frequencies of the distortion products, thereby opposing their amplification. The ear therefore takes advantage of the direct gating of mechanotransduction channels by mechanical forces, including the nonlinear amplification that this feature affords, and at the same time diminishes the distortions created by these nonlinearities. due to a poor signal-to-noise ratio at these frequencies. Although smaller drops in coherence are also apparent at other distortion frequencies, they are typically less prominent than those at the quadratic distortion frequencies, in agreement with earlier results [START_REF] Jaramillo | Auditory illusions and the single hair cell[END_REF][START_REF] Kozlov | Forces between clustered stereocilia minimize friction in the ear on a subnanometre scale[END_REF]. the hair bundle with a phase difference near zero at all frequencies (Fig. S1B). A complete measurement, which comprised twenty such records, yielded similar coherence and phase spectra (Fig. S1C). These results did not differ significantly from those obtained with the two laser beams superimposed at the same position on the hair bundle (Fig. S1D). The average coherence and phase spectra obtained from five hair bundles confirm the consistency of the results (Fig. S2).

Measurements of coherency from isolated hair cells may be corrupted by whole-cell drift.

We therefore performed on each isolated cell a control measurement in which one laser beam was positioned on the hair bundle and the other on the apical portion of the cell body. If the two signals displayed cross-correlations above the background noise level, the cell was rejected from further analysis. For the five cells included in the analysis, the average coherence between measurements from cell bodies and the associated hair bundles was 0.2 ± 0.1 and the average phase was 0.0 ± 1.7 rad between 100 Hz and 5 kHz. This result is consistent with the expectation for two independent signals. In contrast, when the two laser beams were positioned on the opposite sides of these hair bundles, the coherence was 0.88 ± 0.07 and the phase was 0.0 ± 0.1 rad (Fig. S2). These values indicate that auditory hair bundles from the tokay gecko move with a high coherence.

Experimental studies of hair-bundle kinematics with high resolution in isolated cells suffer from at least three methodological drawbacks: cell isolation may cause excessive mechanical and metabolic damage; the proximity of a hair bundle to the recording chamber's glass surface may affect the hydrodynamic forces experienced by the stereocilia; and isolated cells may drift with the local hydrodynamic flow. Although drifting could be strongly reduced by covering the recording chamber with adhesive substances such as concanavalin A and Cell-Tak, most cells nevertheless displayed whole-cell motions of several tens of nanometers. Because these movements were large compared with those typical of relative stereociliary motions, they dominated the measurements. When their global motion was negligible, we could use some isolated cells for recordings of hair-bundle motion. However, stimulating isolated cells with a patch pipette attached to their hair bundles displaced the cell bodies too much to allow reliable measurements of the relative stereociliary motion. Because the basilar papilla of the tokay gecko is thin, with only a dozen hair cells abreast, we could not fold it to provide optical access to laterally protruding hair bundles without damaging the cells. For these reasons, we report in the main text the results from our experiments with the bullfrog's sacculus, which has the advantage of being mechanically stable and metabolically robust: we occasionally observed hair bundles oscillating spontaneously for as long as 20 hours in vitro.

Choice of the multitaper spectral-analysis technique

The structure of the data was investigated through linear frequency-domain representations using the multitaper spectral-estimation method (Thomson, 1982;Percival and Walden, 1993). This choice is motivated by the observation that standard spectral-estimation methods based on single data windows suffer from two fundamental problems, namely bias and lack of consistency of the so-obtained estimates. The first problem refers to the fact that the estimate of the spectral quantity at a given frequency mixes information from different frequency components of the original signal. The second problem refers to the fact that the variance of the estimate does not decline with an increasing sample size, for the outcome contains as many quantities as there are data values. Data tapering by a single window, which is often used in an attempt to solve the first problem, suffers from variance-efficiency reduction, unequal weighting of the data, and arbitrariness in taper selection [START_REF] Brillinger | The key role of tapering in spectrum estimation[END_REF]Thomson, 1982). In our case, the bias reduction offered by the use of a single window was also inadequate because our records contained a relatively important part of low-frequency signal that was unrelated to the desired observations and that could have contaminated the frequency components of interest. To solve the second problem, a convolution product in the frequency domain could be used to smooth the desired estimates, but this operation relies on the assumption that the output spectral quantities are smooth. In our case the spectral power was concentrated at specific frequencies and this condition was unsatisfied.

Details of the multitaper spectral analysis

Any stationary stochastic process € X(t) sampled at a rate (2)

In actual experiments, however, we can observe only a specific realization

€ x(t) of € X(t)
and can do so only over the finite time window € T = Nτ . The observed time series

€ ˜ x (t) has a Fourier transform € ˜ x ( f ) that is related to the Fourier transform € x( f ) of the infinite time series € x(t) by € ˜ x ( f ) = x(t j )e -2πif t j j=1 N ∑ = K( f -f ',N)x( f ')df ' -1/(2τ ) 1/(2τ ) ∫ , (3)
in which

€ t j = jτ and € K( f ) = e -πif (N +1) sin(Nπf ) sin(πf )       . (4)
For a stationary stochastic process the spectrum can be estimated as

€ ˜ x ( f ) 2
, the squared Fourier transform of the data series. However, The resultant over-fitting problem corresponds to the second problem mentioned above, namely lack of consistency.

€ ˜ x ( f ) is not equal to € x( f ) but
The multitaper spectral-estimation method provides an elegant solution to both problems (Thomson, 1982;Percival and Walden, 1993). In this approach, the data are multiplied not by a single window but rather by a set of € K optimally chosen data tapers € w k (t). The power spectrum is then estimated as

€ S MT ( f ) = 1 K ˜ x k ( f ) 2 k=1 K ∑ , ( 5
) in which € ˜ x k ( f ) = w k (t j )x(t j )e -2πif t j j=1 N ∑ . ( 6
)
The optimal choice of the taper functions requires that they be mutually orthogonal, providing € K independent spectral estimates, and that they possess maximal spectral concentration, yielding the greatest relative power over the frequency bandwidth € 2W [START_REF] Mitra | Analysis of dynamic brain imaging data[END_REF]. The spectral concentration value is quantified by [ ]

€ λ k (N,W ) = U k ( f ) 2 df -W W ∫ U k ( f ) 2 df -1/(2τ ) 1/(2τ ) ∫ , (7)
π (t j -t j' ) w(t j' ) j'=1 N ∑ = λw(t j ); (8)
€ w k (t) corresponds to the associated eigenvector under appropriate normalization.

These optimal taper functions have the remarkable property that € 2NWτ of their eigenvalues are approximately equal to one, whereas the remainder decay sharply to zero (Slepian and Pollak, 1961;[START_REF] Slepian | Prolate spheroidal wave functions, Fourier analysis, and uncertainty -V: The discrete case[END_REF]. As a result, the spectrum is convolved with a window that is as close as possible to a rectangular shape, yielding a power-spectral estimate that results at each frequency € f 0 from an equally weighted average of contributions across the frequency band

€ f 0 -W , f 0 + W [ ] .
At the same time, leakage to and from frequencies outside this frequency band is as small as possible.

We chose as the final spectral estimate an equally weighted average of the different tapered spectra. Although more sophisticated techniques of averaging can be implemented with both frequency-and data-dependent weighting (Thomson, 1982;Percival and Walden, 1993;[START_REF] Mitra | Analysis of dynamic brain imaging data[END_REF], these approaches significantly affect the results only when tapers with poor spectral concentration are used. Because we could average over several independent records for each measurement, we did not need to include a large number of tapers to obtain consistent estimates. We could therefore restrict ourselves to tapers with high concentration properties, rendering adaptive-weighting techniques superfluous.

Jackknife error estimation

The multitaper method also permits calculation of the variance of an estimated spectrum by means of jackknifing (Thomson and Chave, 1991). Although this method is applicable to relatively complicated data, it is largely free of distributional assumptions and hence highly reliable. Moreover, the jackknife variance always exceeds the true variance, so the estimates are conservative [START_REF] Efron | The jackknife estimate of variance[END_REF].

The jackknife approach was implemented as follows. Let

( ) = P -1 P ˆ θ \ i -θ ∩ ( ) 2 i=1 P ∑ , (9)
in which

€ θ ∩ = 1 P ˆ θ \ i i=1 P ∑ . (10)
These estimates require the use of transformations prior to jackknifing when the statistics is bounded or its distribution is strongly non-Gaussian, for example strongly asymmetrical about the mean value. We used the logarithmic transformation for jackknifing power spectra and the inverse hyperbolic transformation for jackknifing coherences (Thomson and Chave, 1991).

In power-spectral estimation, a logarithmic transformation stabilizes the distribution into a more symmetric one. Taking (1α) confidence interval for the power spectrum was given by € ˆ S e -t P-1 (1-α / 2) ˆ σ < S ≤ ˆ S e t P-1 (1-α / 2) ˆ σ . ( 14)

Coherence and phase estimation

To jackknife coherence and phase spectral estimates, we assumed the availability of (15) for € j = 1,2,...,P , plus the standard estimate with nothing omitted (Thomson and Chave, 1991).

From these, we transformed to the almost normal variates

€ Q \ j = 2P -2 tanh -1 ˆ c \ j ( ) (16)
and obtained estimates and tolerances of the coherence spectrum.

Delete-one phase factors were constructed as (18)

€ e \ j = ˆ c \ j ˆ c \ j , ( 17
This approach provided an estimate of the phase variance as

€ V ˆ φ { } = 2 P -1 ( ) ⋅ 1-φ ∩ ( ) , (19)
in which € φ ∩ = arg e \ j { } , an application of the jackknife method to standard phase statistics (Fisher, 1993). In particular, the estimate took periodicity into account and was equivalent to

K

  ≤ 2NWτ and that the spectral power of the tapers was sufficiently concentrated in the frequency band

FIGURESFigure 1 :

 1 FIGURES

Figure 2 :

 2 Figure 2: Spectral power, coherence and phase at the distortion frequencies. The average values of the spectral-power density (PSD), coherence and phase for six stimulated cells are shown in red and the standard deviations in blue. Because of the use of a logarithmic scale for the spectral-power density, the values corresponding to the mean minus a standard deviation are not shown in this case. Stereociliary movements at the frequencies of stimulation, f 1 = 90 Hz and f 2 = 115 Hz, display a coherence across the hair bundle close to one and a phase close to zero; at those frequencies the bundle moves as a unit. In contrast, large phase lags and coherence drops are apparent at the distortion frequencies 2•f 1 = 180 Hz, f 2 + f 1 = 205 Hz, and 2•f 2 = 230 Hz. The relatively large associated power-spectral densities confirm that the observed phase lags are not

Figure 3 :

 3 Figure 3: Coherence along the two perpendicular axes of hair-bundle movement. The average coherence values for movement in the orthogonal plane, which are shown in black,exceed those for movement in the plane of mechanosensitivity, which are portrayed in red. The associated standard deviations are displayed in grey and pink, respectively. In both configurations the mean phase was around zero, but the standard deviation between 100 Hz and 5 kHz was smaller for movement in the orthogonal plane than for movement in the plane of mechanosensitivity-0.04 rad versus 0.16 rad-a result that accords with the greater coherence in the former configuration.

  2 ) are statistically uncorrelated. The second moment of this function defines the power spectrum€ S( f ) of the process, € S( f )df = E dX( f ) 2 { } .

  is related to it by a convolution product that mixes information originating in different frequency channels (Equation3); this corresponds to the first problem of bias mentioned above. Moreover,

  f ) is the Fourier transform of the sequence € w k (t) and € τ is the sampling rate. It can be shown that € λ k (N,W ) is the € k th eigenvalue of the eigenvector relation € sin 2πW (t j -t j' )

  degrees of freedom. The approximate €

  f ). We defined the delete-one estimates of the coherency as

  = arg e \ j { } for small phase dispersion.

Figure S1 :

 S1 Figure S1: Time series and coherency spectra from an auditory hair bundle. (A) Simultaneous recordings with two interferometric beams show highly similar patterns of motion at the opposite edges of a gecko's hair bundle. (B) The coherence and phase spectra for the records in panel (A) are shown in red, together with their associated 95 % confidence intervals in blue. (C) Averaging the coherence and phase spectra from 20 records for the opposite edges of the same hair bundle greatly reduces the experimental uncertainty. The confidence intervals were obtained by jackknifing the entire set of records. (D) The average spectra from 20 records with the two beams positioned at an identical position in the same hair bundle closely resemble those in panel (C).

Figure S2 :

 S2 Figure S2: Coherence and phase spectra from auditory hair bundles. The figure displays the average values (red) and the associated standard deviations (blue) of the coherence and phase spectra for recordings from the opposite edges of five hair bundles from the gecko's basilar papilla.

Table 1 :

 1 Coherence and phase values at the stimulation and principal distortion-product frequencies.

	Frequency (Hz)	Coherence ± SD	Phase (radians) ± SD
	90 (f 1 )	0.99923 ± 0.00051	-0.0230 ± 0.0340
	115 (f 2 )	0.99922 ± 0.00050	-0.0272 ± 0.0478
	180 (2•f 1 )	0.32	± 0.15	-0.97 ± 2.46
	205 (f 1 + f 2 )	0.51	± 0.19	-1.65 ± 2.08
	230 (2•f 2 )	0.42	± 0.20	-0.54 ± 1.08

ACKNOWLEDGMENTS: We thank B. Fabella for technical support and programming and O. Ahmad, D. Andor, J. Baumgart, and M. O. Magnasco for discussions. This investigation was funded by grant DC000241 from the National Institutes of Health. A. S. K. was supported by Howard Hughes Medical Institute, of which A. J. Hudspeth is an Investigator.

AUTHOR CONTRIBUTIONS: A. S. K. and A. J. Hudspeth conceived the project; A. S. K. and A. J. Hinterwirth constructed the interferometer; A. S. K. performed the experiments; T. R. implemented the data analysis; A. S. K. and T. R. analyzed the data and wrote the manuscript; A. J. Hinterwirth and A. J. Hudspeth edited the manuscript. The experiments were conducted at The Rockefeller University.

SUPPLEMENTARY INFORMATION

Relative stereociliary motion in a hair bundle opposes amplification at distortion frequencies Andrei S. Kozlov, 1 Thomas Risler, 2,3,4 Armin J. Hinterwirth, 1 and A. J. The experiments presented in the main text were performed on hair cells from the bullfrog's sacculus, an organ responsive to frequencies around 100 Hz [START_REF] Lewis | Tuning in the bullfrog ear[END_REF]. We wished to test the generality of our previous conclusion that hair cells display high coherence (Kozlov et al., 2007) in auditory hair bundles tuned to higher frequencies. For this purpose we recorded from hair bundles of the basilar papilla in the tokay gecko (Gekko gecko), a lizard with a well developed sense of hearing that extends to about 7 kHz [START_REF] Manley | Frequency response of the ear of the Tokay gecko[END_REF].

Cochleae dissected from geckos [START_REF] Chiappe | The structural and functional differentiation of hair cells in a lizard's basilar papilla suggests an operational principle of amniote cochleas[END_REF] were maintained in oxygenated saline solution comprising 170 mM NaCl, 2 mM KCl, 1 mM CaCl 2 , 10 mM D-glucose, and 5 mM HEPES at pH 7.3. After a 30-60 min digestion of basilar papillae at room temperature in 1 mg/ml collagenase (type XI, Sigma Chemical Co.), hair cells were mechanically isolated with an eyelash and allowed to settle onto the bottom of an experimental chamber coated with either concanavalin A (Sigma) or Cell-Tak (BD Biosciences) to promote cellular adhesion.

A typical pair of time series obtained from the opposite edges of a gecko's hair bundle demonstrated fluctuations with a root-mean square (RMS) magnitude of 3.5 ± 0.5 nm (mean ± standard deviation; Fig. S1A). The associated spectra showed a high coherence across


Patrick Maillé
email: patrick.maille@imt.fr

Bruno Tuffin
email: bruno.tuffin@inria.fr

How does imposing free roaming in EU impact users and ISPs' relations?

Starting in June 2017, roaming within Europe will become free for mobile phone users, as an enforcement of the European parliament. The goal being to obtain a unified digital market: this seems at first sight to largely benefit to users, who will not have to worry about their usage abroad. We design in this paper a model analyzing the impact of free roaming enforcement on users and ISPs' pricing strategies, with respect to paid roaming to verify this assumption. We particularly focus on the strategies on transit payments between ISPs in different countries. We highlight that scrutiny is required since, depending on parameters, consumer surplus or subscription penetration are not necessarily maximized if free roaming is enforced.

1 https://ec.europa.eu/digital-single-market/en/news/new-rules-roaming-charges-and-open-internet

Introduction

In traditional mobile network charging plans, roaming in a foreign country implies additional charges, typically volume based or cap based. But in October 2015 the European parliament has decided to forbid roaming charges, starting June 15, 2017, in order to have a unified digital market 1 . Users then just need to subscribe to a plan in their member state, determined as the place where they spend most of their time. As a transitory phase, the roaming charges have already been capped since April 2016.

This free roaming rule does seem attractive at first sight, with mobile users not having to worry about their data consumption while travelling all over Europe. But it may lead to negative effects that we wish to investigate here: i) Despite being appealing, is there a risk to create a price increase by which "poor" domestic-only users would pay for "rich" roaming users? ii) What is the impact of free roaming on transit prices between ISPs in different countries, and then on subscription prices? iii) Is it needed to regulate the way transit prices are decided, or can we leave it open to ISPs (through negotiation or a non-cooperative game)?

In order to answer those questions, we design a model representing the relations between ISPs in two different countries. Each ISP decides the flat rate price for its domestic users and the transit price (for traffic from the ISP in the foreign country) charged to the counterpart ISP. To make comparisons with the case of paid roaming, we will also add the possibility of a volume-based fee for roaming usage, charged to its own users for data consumption abroad.

The two cases, free and paid roaming will be analyzed thanks to noncooperative game theory [START_REF] Maillé | Telecommunication Network Economics: From Theory to Applications[END_REF][START_REF] Osborne | A Course in Game theory[END_REF], since the decision of an ISP will impact the output (the revenue) of the distant ISP from the transit fee and the number of roamers. We will show that scrutiny is required since some a priori counterintuitive results come out. We will illustrate among other things that consumers do not always benefit from free roaming: consumer surplus, representing the aggregated net benefit from users, might even be larger with paid roaming.Due to the effect of prices on subscription rates, free roaming might also lead to less subscriptions.

To our knowledge, a very limited amount of research work exists on the analysis of the enforcement of free roaming, and even on roaming in general. We can cite [START_REF] Stühmeier | Roaming and investments in the mobile internet market[END_REF], but it is rather focusing on national roaming of customers and cooperation among operators. Recently, we have designed and analyzed in [START_REF] Maillé | Enforcing free roaming among UE countries: an economic analysis[END_REF] a model looking at a single ISP optimizing its pricing strategies, both in case of paid and free roaming, and comparing them. Our model makes use of the same representation of users' preferences, but extends it in the sense that it additionally investigates the impact of relations between ISPs in the different countries based on the negotiated transit prices, which can influence the domestic decisions and induce a new level of decisions.

The remainder of this paper is organized as follows. Section 2 presents our model of user preferences and operator revenues as well as the different levels of decisions. In Section 3, we recall the results on users' choices in terms of subscription and data usage depending on prices, as obtained in [START_REF] Maillé | Enforcing free roaming among UE countries: an economic analysis[END_REF]. Section 4 provides some theoretical results on the users' pricing game between ISPs, while Section 5 analyzes transit prices being decided either from a non-cooperative game, through a negotiation (using the so-called Nash bargaining solution [START_REF] Osborne | A Course in Game theory[END_REF]), or by a regulator in order to maximize consumer surplus. Section 6 then compares the outcomes of the combinations of scenarios to determine what would be the best strategy for all actors and discusses the UE parliament decision. Section 7 particularly focuses on the impact of decisions on each individual of the population for an arbitrarily chosen scenario. Finally, we conclude in Section 8.

Model

We consider two countries for simplicity, differentiated by indices 1 and 2, but more countries could be considered as well. According to the rules/recommendations2 , citizens of a country have to subscribe to the/an ISP in their own country to avoid an unfair competition due to different local constraints. We thus avoid the possibility of a citizen of a given country subscribing in another and less expensive country and consuming freely in its own country.

Users' representation

In each Country i ∈ {1, 2}, users have heterogeneous willingness-to-pay values (in monetary units per month) for a flat-rate domestic subscription; we denote that value for a particular user by θ i , and assume that the distribution of θ i over the market is known to the operator. The cumulative distribution function of θ i is denoted by F i and will help us to represent two countries with different GDP per capita, a situation for which free roaming could have a more important impact. In all our numerical analysis, we will assume that θ i is exponentially distributed with rate λ i . Then the smaller λ i , the larger the average wealth. There is a total mass (population) m i of potential users in Country i.

But users are not only interested in a domestic usage of their mobile phone, as they can in addition be willing to consume data while traveling abroad (that is, while roaming). Here too there is some heterogeneity in the willingnessto-pay for data while roaming, but it is reasonable to assume a positive correlation between the willingness-to-pay θ i for the domestic data plan, and the willingness-to-pay function (in terms of volume) for roaming data. Let us denote this willingness-to-pay for roaming data by r θi (x) for a consumed volume x over a month. All over the paper, we are going to consider the commonly adopted following form of r θi , see [START_REF] Cho | Pricing for mobile data services considering service evolution and change of user heterogeneity[END_REF][START_REF] Maillé | Users facing volume-based and flat-rate-based charging schemes at the same time[END_REF]:

Assumption A For any value of θ, the function r θ is such that

r θ (x) = θx -α x 2 2 if x ≤ θ α θ 2 2α otherwise.
with α a fixed parameter. The marginal valuation function r θ then has the simple expression

r θ (x) = [θ -αx] +
where [y] + := max(y, 0). This specific form verifies expected properties: for any value of θ i , it is nonnegative, null at 0, non-decreasing, concave and bounded. Moreover, it verifies that users with high willingness-to-pay θ i for domestic usage are also willing to pay more for roaming data usage than users with low θ i (the so-called "positive correlation" mentioned above).

The ISP in Country i is charging a flat-rate p i to its subscribers for a domestic usage. We will compare free and paid roaming, so let us define c i as the fixed per-volume price for roaming data; free roaming will just mean c i = 0.

Given those notations, the net utility of a user with domestic willingness-topay θ i in Country i and consuming a volume v i (θ i ) of roaming data per month (on average), is

U i (θ i ) := θ i -p i + r θi (v i (θ i )) -c i v i (θ i ), (1)
where θ i -p i stands for the net utility for domestic usage and

r θi (v i (θ i ))-c i v i (θ i )
is the net utility for roaming. That form defines the behavior of any user, subscribing if getting a non-negative net utility. Let

D i := {θ i : U i (θ i ) ≥ 0}
be the set of users θ i subscribing to the offer of ISP i, meaning that its associated demand-the mass of subscribers to ISP i-is

d i := m i Di dF i (θ i ).

ISPs' revenues and Consumer Surplus

Each ISP i chooses its subscription price p i , and (in case of paid roaming) the roaming volume-based price c i charged to users, but can also charge a transit price t i to the counterpart ISP for the traffic from foreign users using its network.

There are thus three decision variables for each ISP. We here potentially consider that an ISP may charge more its own users than what it has to pay itself for roaming data, that is, c i can be larger than t j with j = i.

From those assumptions, the revenue (that is, the utility) that ISP i tries to maximize is

R i = d i p i + (c i -t j )m i Di v i (θ i )dF i (θ i ) + t i m j Dj v j (θ j )dF j (θ j ). (2)
Another measure of interest, especially from a European regulator (the BEREC) point of view seeking to maximize user welfare, is the consumer surplus representing the aggregated net utility of users over all countries:

CS = i m i [U i (θ i )] + dF i (θ i ) = i m i Di U i (θ i )dF i (θ i ).
(3)

Scenarios and hierarchy of decisions

Several scenarios can be considered for the decision of transit prices t 1 and t 2 , while prices charges to users are always chosen to maximize the ISPs' revenues.

In all cases, we will compare the free roaming situation where c 1 = c 2 = 0 with the paid roaming.

The scenarios for the decision on the transit prices are the following:

1. t 1 and t 2 decided by a regulator to maximize the consumer surplus;

2. t 1 and t 2 decided as a Nash Bargaining solution [START_REF] Osborne | A Course in Game theory[END_REF] under the threat that the regulator will choose of no agreement is found;

3. (t 1 , t 2 ) a Nash equilibrium as the result of a non-cooperative game.

There is a natural order of decisions:

1. The transit price(s) t i ∀i are first decided according to one of the three above possibilities;

2. Then prices are defined by the ISPs: p i ∀i, as well as the c i s in case of paid roaming;

3. Based on all those values users decide to subscribe or not, defining the domains D i ∀i.

But decisions are taken anticipatively, meaning that decisions at a given level will be made anticipating the decisions afterwards. The game is thus analyzed by so-called backward induction [START_REF] Maillé | Telecommunication Network Economics: From Theory to Applications[END_REF]. We therefore first compute the domains D i over θ i of users choosing to subscribe in terms of all other parameters, then the ISP prices using those D i s for any choice of t i s, and then the t i s using the anticipated (previously computed) values.

User decisions

Since the form of the user utility function described in Subsection 2.1 is taken from the literature, the derivations of user choices have already been studied. Based on utility function [START_REF] Cho | Pricing for mobile data services considering service evolution and change of user heterogeneity[END_REF], it has been shown in [START_REF] Maillé | Enforcing free roaming among UE countries: an economic analysis[END_REF] in the context of a single ISP and considering a single country, but the results remain valid for our model, that

• Under Assumption A, by a simple derivation with respect to v i , the con-

sumed volume is for a user θ i is v * i (θ i ) = θi-ci αi +
, so the user subscribes if and only if θ + (θ-c) 2 2α 1l {θ>c} > p. • For any price pair (p i , c i ) the users who subscribe are those with θ i > θi , where θi = min p i , c i -

α i + α 2 i + 2α i [p i -c i ] + , (4)
leading to

D i = ( θi , ∞).
In words, users with valuations above the threshold θi will subscribe.

On the pricing game between ISPs

For fixed values of t 1 and t 2 , what is the output of the pricing game between ISPs? In their choice, ISPs anticipate the reaction of users, hence know the values θ1 and θ2 for any profile of subscription (and roaming usage, if any) prices.

We have a non-cooperative game because the revenue (2) of one ISP depends on the prices of the other ISP through the demand for roaming data from this other ISP. However, for free and paid roaming, we have the following result which simplifies the analysis.

Proposition 1 The determination of revenue-optimizing prices (p i , c i ) of ISPs is independent of the pricing strategy (p j , c j ) of the other ISP j.

Proof: It actually comes from the expression (2) for the revenue of ISP i. The best response of ISP i is obtained by differentiating (2) with respect to p i and c i (or just with respect to p i if c i = 0 in the case of free roaming). Revenue R i is made of three components:

• d i p i which depends only on decision variables of ISP i;

• (c i -t j )m i Di v i (θ i )dF i (θ i ) which depends on ISP j only through t j , but
that is a constant at this level of the game;

• t i m j Dj v j (θ j )dF j (θ j ) which does not depend on (p i , c i ), and therefore is a constant for the optimization at this level of the game.

As a consequence, we get the proposition.

For free roaming, we have the expression of revenue of ISP i

R i = d i p i -t j m i Di v i (θ i )dF i (θ i ) + t i m j Dj v j (θ j )dF j (θ j ). ( 5
)
With exponential distributions for θ 1 and θ 2 and the expressions of θ1 and θ2 given in previous section, it gives for free roaming

R i = m i e -λi θi p i -t j mi αi e -λi θi θi + 1 λi e -λi θi +t i mj αj
e -λj θj θj + 1 λj e -λj θj , and, for paid roaming,

R i = m i e -λi θi p i -(c i -t j ) mi αi e -λi θi [ θi -c i ] + + e -λ i c i λi e -λi[ θi-ci] + +t i mj αj e -λj θj [ θj -c j ] + + e -λ j c j λj e -λj [ θj -cj ] + .
Computing the revenue-maximizing prices analytically seems intractable, but it can easily be done numerically. For example when

α 1 = α 2 = 1, λ 1 = 1, λ 2 = 1.5, m 1 = 1,

On the determination of transit prices

Again, we are going to consider as much as possible paid and free roaming simultaneously, since free roaming can be summarized by taking c 1 = c 2 = 0.

Transit price decided by a regulator

As a first possibility, we can let a regulator (like BEREC, the European one) decide the transit prices t 1 and t 2 . A goal would be then to maximize the consumer surplus

CS = i m i Di U i (θ i )dF i (θ i ) = i m i Di θ i -p i + ([θ i -c i ] + ) 2 /(2α i ) dF i (θ i ).
For free roaming, it reduces to CS = i m i Di θ i -p i + θ 2 i /(2α i ) dF i (θ i ). Remark first that the expression of CS does not directly depend on the t i s, but it does through the chosen p i s, which depend on the t i s from the second term in [START_REF] Maillé | Telecommunication Network Economics: From Theory to Applications[END_REF].

Through extensive computations, we have found in all our experiments that CS is maximized at (t 1 = 0, t 2 = 0). We do not claim that it is always the case, but it seems to happen, at least with our parameters. Remark to understand what could drive to this result that a sufficient condition for CS to be maximized at (t 1 = 0, t 2 = 0) would be that the p i s and c i s are non-decreasing with the t i s since D i then decreases. The increase of p i in terms of t j depends on the respective evolution of the derivatives (in terms of p i ) of d i p i and m i Di v i (θ i )dF i (θ i ), but a proof or simple condition for seems hard to find and is out of the scope of this paper.

Transit prices as a Nash bargaining solution

Another solution is to consider the so-called Nash bargaining solution [START_REF] Nash | Two-person cooperative games[END_REF], output of a negotiation between ISPs: each ISP independently chooses a set of acceptable transit prices, and if the intersection of those sets is non-empty, the transit price is arbitrarily taken in the intersection. If ISPs do not agree with this, a threat is executed. A reasonable threat here is to take zero transit prices because if no agreement is found, the regulator can impose that no transit fee be applied. This negotiation scheme has in general several equilibria, but in [START_REF] Nash | Two-person cooperative games[END_REF] it is stated that the most likely outcome is the one maximizing the product of the utilities minus the utility at the threat situation. Formally, if R i (t 1 , t 2 ) is the revenue if ISP i where the roles of t 1 and t 2 are highlighted (with the corresponding equilibrium subscription fees), then the Nash bargaining solution (t B 1 , t B 2 ) is defined as:

(t B 1 , t B 2 ) ∈ arg max (t1,t2) R 1 (t 1 , t 2 ) -R 1 (0, 0) R 2 (t 1 , t 2 ) -R 2 (0, 0) , 0 0.1 0.2 0.3 0.4 0.5 0.4 0.6 0.8
Free transit

t 2 Revenues R 1 (0.1, t 2 ) R 2 (0.1, t 2 ) R 1 (0, 0) R 2 (0, 0)
Figure 1: Free-roaming ISP revenues for t 1 = 0.1, with

α 1 = α 2 = 1, λ 1 = 1, λ 2 = 1.5, m 1 = 1, m 2 = 2.
Subscription prices are set through a non-cooperative game among ISPs.

with the constraint that both terms are non-negative, because otherwise the ISPs prefer the no-transit-fee solution.

Again, during all our numerical investigations, the Nash bargaining solution was also at (0, 0). The reason is that with combinations of (t 1 , t 2 ) at least one of the ISPs get a smaller revenue than when it is (0, 0). We can illustrate this in Figure 1, where we plot for t 1 = 0.1 the revenues in terms of t 2 and the threat values R 1 (0, 0), R 2 (0, 0), in the case of free roaming (that is, c 1 = c 2 = 0).

Transit prices from a game between ISPs

Finally, the couple (t 1 , t 2 ) can be the output (a Nash equilibrium [START_REF] Osborne | A Course in Game theory[END_REF]) of a noncooperative game between ISPs, since the revenues depend on the transit price of the other ISP. But similarly to the users' pricing game, we have the following result saying that even if there is a dependence of revenues, the best-response transit price of an ISP does not depend on the strategy of the other ISP.

Proposition 2

The revenue-optimizing transit price t i of ISP i is independent of the strategy t j of ISP j (even if its revenue is not).

Proof: First remark that in the proof of Proposition 1, p i (and c i if paid roaming is considered), and therefore D i and d i depend on t j but not on t i . Therefore, examining the expression (2) we observe that

• d i p i + (c i -t j )m i Di v i (θ i )dF i (θ i )
depends only on t j and not on t i ;

• t i m j Dj v j (θ j )dF j (θ j ) is the component that depends on t i and not on t j since D j depends on t i and not t j .

As a consequence, we get the proposition.

Comparison and discussion

In this section, we compare the outputs of the various scenarios, with transit prices chosen by the regulator, as a Nash bargaining solution or through a noncooperative game, each time for free and paid roaming.

As discussed above, transit prices chosen by the regulator or as a Nash bargaining solution always give (t 1 , t 2 ) = (0, 0) and are therefore merged into one case.

In the first subsection, we discuss the output when varying λ 1 , the other parameters being fixed. We also studied the variations in terms of other parameters but it did not provide any meaningful information; for example varying m 1 did not bring any significant impact on the various considered metrics. In the last subsection, we discuss for specific parameters the impact on each individual θ i of the various strategies: free or paid roaming, combined with transit prices decided from a non-cooperative game or chosen as (0, 0).

Impact of wealth parameter λ 1 in Country 1

Let us fix (arbitrarily) the other model parameters, to α 1 = α 2 = 1, m 1 = 1, m 2 = 2 and λ 2 = 1. We investigate the impact of λ 1 for the different strategies on CS in Figure 2, on p i in Figure 3, on c i (in the paid roaming cases) in Figure 4, on transit prices in Figure 5, on revenues of ISPs in Figure 6, on the thresholds θi in Figure 7, and on the proportions to subscribe Fi ( θi ) in each country in Figure 8.

Discussing the impact of the density of valuation in Country 1 on CS, we can remark that the larger λ 1 , that is the smaller the valuations in distribution, then the smaller the CS at equilibrium. The situation with null transit prices (that is, decided by Nash bargaining or the regulator) and free roaming is (here) always giving the largest CS among all possibilities, hence should be the preferred situation for a regulator. Null transit prices always give a better CS that when transit prices are decided through a non-cooperative game, be it with free or paid roaming. Finally, focusing on paid and free roaming, when transit prices (t 1 , t 2 ) are chosen through a non-cooperative game, there is no clear better-performing scheme, in terms of consumer surplus, between free and paid roaming. It means that enforcing free roaming could actually be a false good idea depending on how prices are decided if transit price decisions are let to ISPs.

Looking at prices, wealth (λ 1 ) in a country does not impact the pricing strategies (p 2 , c 2 ) in the other country. We can see on Figure 3 that the smaller λ 1 is, i.e., the larger the number of users with high valuation is, the larger the subscription price p 1 and roaming prices c 1 are, because users are more likely to accept high prices. Also free roaming induces a higher price because ISP 1 cannot get more money based on volume. Moreover subscription prices with free roaming are higher at Nash-equilibrium transit prices than with (0,0) to compensate the transit cost, but quite surprisingly with paid roaming, they are the same; actually, as illustrated on Figure 4, transit prices are compensated by roaming charges. The level of wealth in a country is illustrated in Figure 5, and as proved earlier, to impact the transit price of the other country (because it can attract more roaming money) but not the domestic transit prices. Interestingly also, there is no strict dominance of transit prices between free and paid roaming: when λ 1 is small, i.e., users are wealthier, the transit price from the other country is larger with paid roaming than with free roaming, but it is the opposite when λ 1 is large; this can probably be related to the increased roaming charge c i which increases with wealth, inducing a possibility to increase the transit prices (indeed, the two values are highly positively correlated).

Revenues in terms of λ 1 are shown in Figure 6. They are decreasing with wealth, which is coherent; only R 2 is independent of the wealth in Country 1 when transit prices are (0, 0), for free or paid roaming. It is also interesting to see that imposing null transit prices may lead to larger revenues for ISPs with respect to prices defined through a game, the preference depending on the level of wealth in the country(ies). But for a fixed strategy for the definition of transit prices, ISP revenues seem always larger with paid roaming than with free roaming.

Looking more closely at the evolution of subscription thresholds θi in Figure 7, the wealth distribution in Country 1 still does not impact Country 2, because prices are not impacted there. On the other hand, θ1 depends directly on p 1 (and c 1 ), hence decreases with λ 1 , i.e., when users are less wealthy. For the paid roaming cases, the curves with zero or Nash transit prices are superimposed: there is no dependence on the choice of (t 1 , t 2 ).

Figure 8 displays the proportions of subscribers in the different cases; again wealth in a country does not impact subscriptions in the other country. Wealth has no impact on the proportion in the case of paid roaming, which is actually not true for free roaming (ex: the proportion of subscribers in Country 1 in the free and regulated case is increasing as λ 1 increases, because of the price decrease). Paid roaming is actually more interesting if the proportion of subscribers is what we are focusing on.

7 Users' utility in terms of θ

We now fix the parameters to α 1 = α 2 = 1, m 1 = 1, m 2 = 2, λ 2 = 1 and λ 1 = 2. We use the prices determined for the four strategies (free or paid roaming, null transit prices or determined through a Nash equilibrium), and display in Figure 9 the individual user utility as a function of θ 1 . We can see that for a large θ 1 the free options will always be preferred, which is not true when θ 1 is small. Hence the consumer surplus should depend on the distribution of wealth, i.e., of θ 1 . The gap between the two options with transit prices decided through a non-cooperative game shows why the paid option is prefered in this case (given that a small θ 1 is much more likely to occur). Also we see why the consumer surplus is larger with the free option with transit prices (0, 0).

Conclusions

  m 2 = 2 and t 1 = t 2 = 0.1, we get as revenue-optimizing prices (p 1 , c 1 ) = (1, 1.1) and (p 2 , c 2 ) = (0.67, 0.765).

Figure 2 :

 2 Figure 2: Consumer Surplus in terms of λ 1 , for free and paid roaming, with different transit prices.

1 (

 1 free roaming, (0, 0) transit) p 1 (free roaming, NE transit) p 1 (paid roaming, (0, 0) transit) p 1 (paid roaming, NE transit) p 2 (free roaming, (0, 0) transit) p 2 (free roaming, NE transit) p 2 (paid roaming, (0, 0) transit) p 2 (paid roaming, NE transit)

Figure 3 : 1 ,Figure 4 :

 314 Figure 3: Prices p i in terms of λ 1 , for free and paid roaming, and different transit prices.

1

 1 Nash equilibrium transit prices t 1 , paid roaming t 2 , paid roaming t 1 , free roaming t 2 , free roaming

Figure 5 :R 1 ,

 51 Figure 5: Nash equilibrium transit prices in terms of λ 1

Figure 6 :

 6 Figure 6: Revenues in terms of λ 1 , for free and paid roaming, and different transit prices.

Figure 7 : 1

 71 Figure 7: Subscription threshold θi in terms of λ 1

Figure 8 :U 1 ,Figure 9 :

 819 Figure 8: Proportion of users to subscribe in terms of λ 1

see again https://ec.europa.eu/digital-single-market/en/news/ new-rules-roaming-charges-and-open-internet

We have provided in this paper a model describing the interactions of ISPs in different countries, and subscription choices of users potentially subject to roaming. The goal was to analyze whether or not the decision to impose free roaming in EU in July 2017 is good for users -which is expected-and ISPs -which is less expected-.

We have illustrated some a priori counter-intuitive results: free roaming is not necessarily better for users than paid roaming (in particular, if transit prices are decided by ISPs through a non-cooperative game), and ISPs' revenues are not necessarily larger if defined by the ISPs themselves. It shows that scrutiny is required before taking a decision.

This work can be extended in several ways. First, we may add competition among ISPs in given countries, or different levels of competition to see if this does bring an asymmetry in the relations. Also some of our results are obtained with full generality while others, even if illustrative of the required scrutiny, are from numerical investigations with specific functions; we would look to generalize those observations.


M Corso

A Vannozzi

E Maza

N Vitulo

F Meggio

M Bouzayen

G Valle

C Bonghi

M Lucchin

Transcriptome pathways in leaf and root of grapevine genotypes with contrasting drought tolerance

Keywords: Vitis, water stress, mRNA-seq, flavonoids, stilbenes, WRKY

Most of the world's wine-producing regions are subjected to seasonal drought, and, in the light of the dramatic climate-change events occurring in recent years, the selection of resistant rootstocks is becoming a crucial factor for the development of sustainable agricultural models to ensure optimal grape berry development and ripening. In this study, roots and leaves of 101.14 (drought-susceptible) and M4 (drought-tolerant) rootstocks were sampled in progressive drought and mRNA-seq profiles were evaluated. Physiological characterization indicated that only M4 was able to maintain high leaf transpiration and net assimilation rates under severe stress conditions. Statistical analyses, carried out on mRNA-seq data, highlighted that "treatment" (water stress) and "genotype" (rootstock-genotype) seem to be the main variables explaining differential gene expression in roots and leaves tissues, respectively. Upon water-stress, roots and leaves of the tolerant genotype M4 exhibit a higher induction of stilbenes (i.e., STS) and flavonoids (e.g., CHS, F3H, FLS) biosynthetic genes. Moreover, the higher expression of STS genes in M4 is coupled with an up-regulation of WRKYs transcription factors. STS genes promoter regions, extracted from whole genome of M4 and 101.14, highlighted a higher number of W-BOX cis elements (binding site for WRKYs) in the tolerant genotype.

INTRODUCTION

Most wine-producing regions in the world are subjected to seasonal drought, and, based on the global climate models predicting an increase in aridity in the future (IPCC, 2007), water deficit may became the major limiting factor in wine production and quality. Drought is associated with many morphological and physiological changes in plants, including reduced expansion of aerial organs [START_REF] Cramer | Water and salinity stress in grapevines: early and late changes in transcript and metabolite profiles[END_REF], lack of root growth [START_REF] Sharp | Solute regulation and growth by roots and shoots of water-stressed maize plants[END_REF], decrease in transpiration rate and photosynthesis [START_REF] Chaves | Grapevine under deficit irrigation: hints from physiological and molecular data[END_REF], accumulation of osmolytes [START_REF] Cramer | Water and salinity stress in grapevines: early and late changes in transcript and metabolite profiles[END_REF], activation of detoxifying processes. Moreover, it affected the transcriptional regulation of a large number of genes [START_REF] Cramer | Water and salinity stress in grapevines: early and late changes in transcript and metabolite profiles[END_REF][START_REF] Tillett | Identification of tissuespecific, abiotic stress-responsive gene expression patterns in wine grape (Vitis vinifera L.) based on curation and mining of large-scale EST data sets[END_REF]. Grapevines are well adapted to semi-arid climates such as that of Mediterranean regions and are generally considered as relatively tolerant to water deficit. However, given that a large proportion of vineyards are located in regions where seasonal drought coincides with the grapevine growing season, the combined effect of soil water deficit, air temperature and high evaporative demand is known to limits yield and delays the vintage date [START_REF] Chaves | Grapevine under deficit irrigation: hints from physiological and molecular data[END_REF], with a negative effect on the berries and, consequently, wine quality.

In addition to their well known resistance to phylloxera, rootstocks derived from grapevine American species are known to confer tolerance to a larger range of environmental stresses, such as drought, high salinity and low Fe [START_REF] Corso | Grapevine rootstock effects on abiotic stress tolerance[END_REF].

A biochemical and physiological study of a novel candidate genotype to be used as rootstock in grapevine was performed recently [START_REF] Meggio | Biochemical and physiological responses of two grapevine rootstock genotypes to drought and salt treatments[END_REF]. This genotype, named M4, was selected for its high tolerance to water deficit (WS). In comparison with the 101.14 commercial genotype, M4 ungrafted plants showed a greater capacity to tolerate WS, maintaining photosynthetic activity also under severe stress conditions. Here we report large-scale whole transcriptome analyses performed on leaf and root tissues of both M4 and 101.14 genotypes under the same WS experimental conditions described by [START_REF] Meggio | Biochemical and physiological responses of two grapevine rootstock genotypes to drought and salt treatments[END_REF]. Water deprivation in our experiment was accomplished gradually, mimicking conditions occurring in the field. Another innovative aspect of the study is the comparative approach between a drought tolerant and susceptible genotype with regard to water stress, which has not so far been explored.

MATERIALS AND METHODS

Two-year-old glasshouse-grown plants of the susceptible and tolerant grapevine rootstocks, respectively the widely used 101.14 genotype (V. riparia × V. rupestris) and the experimental M4 ((V. vinifera × V. berlandieri) × V. berlandieri 'Resseguier n.1') were subjected to drought stress. Seventy-two glasshouse-grown plants from each genotype were divided into two groups: plants grown under well-watered conditions (80% of field capacity) and plants grown under water deficit conditions (from 80 to 30% of field capacity). The whole drought experiment lasted 10 days, during which leaf physiological measurements were performed on fully expanded leaves coming from at least six plants for each genotype and considered time point. Leaf transpiration rate (E, mmol H2O m -2 s -1 ) measurement was performed in the following experimental conditions: 600 μmol of photons m -2 s -1 , a CO 2 concentration of 380 μmol mol -1 , 1.5 kPa of vapor pressure deficit (VPD) and a block temperature of 25°C.

As for the mRNA-seq experiment, M4 and 101.14 plants were divided into two replicates (at least six plants for each group) and leaf and roots were sampled at four time points (from T1 to T4 corresponding to 2, 4, 7and 10 Days After Stress Imposition, DASI, respectively). Total RNA was extracted from frozen grapevine tissues of all samples using the "SpectrumTM Plant total RNA Kit" (Sigma) according to manufacturer instructions. All samples were subjected to mRNA-sequencing (mRNA-seq) according to the CRIBI protocol (http://genomics.cribi.unipd.it/main/services/). DEseq R package (http://www.r-project.org/) was used to perform the statistical analyses for discovering differentially expressed genes (DEGs) by using a multi-factorial approach [START_REF] Maza | Comparison of normalization methods for differential gene expression analysis in RNA-Seq experiments: a matter of relative size of studied transcriptomes[END_REF]. DEGs results from the multifactorial analysis (in particular those in common between all three components and only those in common between R and T) were grouped based on ontological categories using Blast2GO software v2.5.0 [START_REF] Götz | High-throughput functional annotation and data mining with the Blast2GO suite[END_REF].

Subsequently, genes belonging to selected categories of GO was carried out "Differential Cluster Analysis" (DCA). Briefly, the DCA consists in i) clusterization, on the basis of expression profiles, of the genes-set related to M4 rootstock; ii) set-up of the corresponding cluster in 101.14; iii) calculation of the correlation values between M4 and 101.14 genes belonging to the same ontological categories; iv) identification of conserved clusters (high correlation) and not (low correlation value) between the two rootstocks.

RESULTS AND DISCUSSION

The study aims to shed light on the genetic determinism underlying tolerance/ susceptibility to water stress of M4 and 101.14 grapevine rootstocks, displaying high and low levels of tolerance, respectively.

The transpiration rate (E) measured in WW conditions was 2.73±0.3 and 2.4±0.2 mmol H 2 O m -2 s -1 for M4 and 101.14 genotypes, respectively, and maintained similar values throughout the experiment. On the other hand, under WS, as drought gradually led to severe stress conditions (around 30% of field capacity), 101.14 plants showed almost complete stomatal closure, whereas M4 plants maintained a transpiration rate of approximately 20% with respect to the control (Figure 1). These data are in accordance with those discussed by [START_REF] Meggio | Biochemical and physiological responses of two grapevine rootstock genotypes to drought and salt treatments[END_REF] that observed a concurrent decrease of net assimilation rate (A n ) and stomatal conductance (g s ) in both genotypes at early stages of WS, but at later time points the two genotypes showed a contrasting physiological response to water stress. Indeed, an almost complete inhibition of both assimilation and transpiration rates was observed in 101.14, while M4 plants maintained a g s value corresponding to 20% of that in the control; this allowed higher transpiration rates (24%) and therefore A n values maintaining an active plant growth (60% of that in control plants). These data indicate that, after a decrease of all physiological parameters occurring in both genotypes at the early stages of drought [START_REF] Meggio | Biochemical and physiological responses of two grapevine rootstock genotypes to drought and salt treatments[END_REF], only the M4 genotype was able to maintain higher E and A n rates in more severe stress condition, demonstrating a much better ability to acclimatize than the susceptible genotype. The comparison of leaf and root mRNA-seq data obtained by multifactorial analyses (Figure 2) indicated that in roots the "treatment" factor is the main variable that affected genes expression (7905 DEGs), whereas in leaves the weight of the genotype (i.e., rootstock) was the most prominent (3794 DEGs). This observation is not surprising, given that the root system is the first organ to perceive water deprivation and actively respond to this stress, and thus the type of treatment represents the main variable influencing gene expression. The opposite is true in the leaves, where the genotype factor appears to have a major effect when compared to the other components. This is in agreement with a previous report [START_REF] Frensch | Primary responses of root and leaf elongation to water deficits in the atmosphere and soil solution[END_REF] that in maize plants undergoing water deprivation, roots and leaves use different strategies to respond to stress with a preferential growth of root system over shoots.

In order to identify specific metabolic pathways differentially affected by drought stress in M4 and 101.14 roots and leaves, the DEGs captured by multifactorial analysis, i.e., the DEGs common to R, T, and P components and those common to rootstocks and treatment components, were associated to their respective GO terms. These represented 4072 and 1152 genes in roots and leaves, respectively. GO terms were grouped into macro-categories. Amongst the macro-categories related to root, "transcription factors" was the one counting the highest number of DEGs, with 307 genes corresponding to 7.5% of all DEGs considered (4072 in total). There were 209 and 223 "secondary metabolism" and "sugars" DEGs, respectively, corresponding to 5.1 and 5.5% of total DEGs. The DEGs identified in WS leaves highlighted a different weight of ontology categories compared to the root tissue. Transcripts related to "sugars" were the most highly represented, with 73 DEGs (6.3%). "Transcription factors" and "secondary metabolism" categories were represented with 56 (4.9%) and 54 (4.7%) genes, respectively. Finally, 'plant hormones', 'antioxidant responses' and 'cell wall' categories were less represented in both roots and leaves. 'Secondary metabolism' ontology category was the most interesting because it is common to both root and leaves, but involved different responses (Figure 3). In particular M4 roots (Figure 3A) showed a strong up-regulation of genes involved in resveratrol biosynthesis (e.g., VvSTS18/24/27) after T2 and an induction of four GST-related genes at T1 and T2-T4, in comparison to what observed for 101.14. In addition to the well-known effect of many biotic and abiotic stresses on the induction of STS genes [START_REF] Versari | Stilbene compounds and stilbene synthase expression during ripening, wilting, and UV treatment in grape cv[END_REF][START_REF] Vannozzi | Genome-wide analysis of the grapevine stilbene synthase multigenic family: genomic organization and expression profiles upon biotic and abiotic stresses[END_REF], it has recently been reported that oxidative stresses also appear to upregulate STS genes in Vitis rotundifolia hairy roots [START_REF] Nopo-Olazabal | Differential induction of antioxidant stilbenoids in hairy roots of Vitis rotundifolia treated with methyl jasmonate and hydrogen peroxide[END_REF]. Differently from what observed for roots, in M4 stressed leaves, transcripts coding for enzymes involved in flavonoids biosynthesis (e.g., CHS, F3H, FLS, LDOX) undergo strong induction at T1 and a strong accumulation at T2 (log2 WS/WW of 2-4) (Figure 3B). The up-regulation of these genes and the accumulation of polyphenols in response to drought in the aerial part of the plant were reported in grapevine and in other species [START_REF] Castellarin | Water deficits accelerate ripening and induce changes in gene expression regulating flavonoid biosynthesis in grape berries[END_REF][START_REF] Ramakrishna | Influence of abiotic stress signals on secondary metabolites in plants[END_REF].

Stilbenes and flavonoids have ROS scavenging activity that protects against oxidative damage and controls ROS levels, which is mandatory for plant survival in the presence of abiotic stresses [START_REF] Brunetti | Flavonoids as antioxidants and developmental regulators: relative significance in plants and humans[END_REF]. Stilbenoids (resveratrol in particular) are powerful defence antioxidant molecules found in several species and their accumulation is particularly high in grapevine [START_REF] Vannozzi | Genome-wide analysis of the grapevine stilbene synthase multigenic family: genomic organization and expression profiles upon biotic and abiotic stresses[END_REF][START_REF] Höll | The R2R3-MYB transcription factors MYB14 and MYB15 regulate stilbene biosynthesis in Vitis vinifera[END_REF][START_REF] Stuart | Resveratrol and its derivatives as phytoalexins[END_REF]. Furthermore, it has been suggested that flavonoids, whose biosynthetic genes are induced only in M4 leaves under WS (Figure 3B), act as antioxidants in plant response to oxidative stresses [START_REF] Brunetti | Flavonoids as antioxidants and developmental regulators: relative significance in plants and humans[END_REF]. Indeed, they protect against oxidative stress due to an excess of excitation energy in the chloroplast by absorbing solar wavelengths [START_REF] Agati | Flavonoids as antioxidants in plants: location and functional significance[END_REF] and environmental perturbations [START_REF] Agati | Flavonoids as antioxidants in plants: location and functional significance[END_REF][START_REF] Brunetti | Flavonoids as antioxidants and developmental regulators: relative significance in plants and humans[END_REF]. Flavonoids may also reduce the activity of 'primary' ROS scavenger enzymes (i.e., SOD and CAT) in the chloroplast [START_REF] Brunetti | Flavonoids as antioxidants and developmental regulators: relative significance in plants and humans[END_REF]. In addition, flavonoids are capable of quenching H2O2 and other free-radicals, thus protecting the chloroplast membrane from oxidative damage by stabilizing membranes containing non-bilayer lipids [START_REF] Agati | Flavonoids as antioxidants in plants: location and functional significance[END_REF]. The data support the hypothesis that in addition to the activation of "primary mechanisms" of ROS scavenging, drought-tolerant Vitis species, such as M4, also induce "secondary mechanisms" leading to the biosynthesis of other types of secondary compounds in roots and leaves.

TFs likely play an important role in the differential regulation of flavonoids/ stilbenoids-related genes in roots and leaves of both M4 and 101.14 genotypes. Stressresponsive TFs interact with cis-elements in the promoter regions of several stress-related genes and thus modulate the expression of target genes resulting in imparting abiotic stress tolerance [START_REF] Agarwal | Transcription factors in plants and ABA dependent and independent abiotic stress signalling[END_REF]. In root, three classes of TFs (WRKY, MYB and NAC) showed major differences in transcript abundance between M4 and 101.14 in response to water deficit. The most represented among these are WRKY genes (Figure 3C) encoding an important class of transcriptional regulators involved in biotic and abiotic stresses (e.g., drought, high salinity, UV). A relatively large group of VvWRKY genes were strongly induced in M4 root at 4 DASI. The co-expression of a large number of WRKYs (VvWRKY24/28/29/ 37/41) and VvSTSs genes (Figure 3A,C), raises the hypothesis that some of these WRKYs are actively involved in the regulation of STSs gene expression in Vitis species. This relationship is further supported by the presence of W-BOX Cis-regulatory elements in the promoter region of M4 and 101.14 VvSTSs (data not shown). It was shown recently that is possible to predict the expression level of a given gene based on a cis-regulatory network obtained from promoter sequences and gene co-expression networks [START_REF] Gao | A genome-wide cis-regulatory element discovery method based on promoter sequences and gene co-expression networks[END_REF]. In this context, the higher expression of VvSTSs genes under WS in M4, can be related to the significantly higher frequency of WBOXs in their promoter regions in comparison to that observed in 101.14 and PN40024 (data not shown).

Fewer TFs-related DEGs were observed in leaf compared to root tissues, and most of these TFs belong to the MYB-family. In this study, all MYBs identified as DEGs were induced at T1 and T2 (Figure 3D) in the tolerant rootstock and their expression paralleled that observed for DEGs coding for flavonoid biosynthetic enzymes (Figure 3B). A strong connection between MYB and flavonoids is well documented (for a review see [START_REF] Czemmel | R2R3 MYB transcription factors: key regulators of the flavonoid biosynthetic pathway in grapevine[END_REF]. R2R3-MYB proteins have been identified as key determinants in regulatory networks controlling the expression of flavonoids-related genes during plant/fruit development and in responses to biotic and abiotic stresses.

CONCLUSIONS

The data provide a comprehensive description of the transcriptomic responses to drought in roots and leaves of two genotypes. In contrast to other studies [START_REF] Cramer | Water and salinity stress in grapevines: early and late changes in transcript and metabolite profiles[END_REF][START_REF] Tillett | Identification of tissuespecific, abiotic stress-responsive gene expression patterns in wine grape (Vitis vinifera L.) based on curation and mining of large-scale EST data sets[END_REF], we did not consider responses to water stress that are common to susceptible and tolerant plants (e.g., genes involved in ABA metabolism or those responsible for ROS production and primary ROS scavenging), but rather focused on genes whose expression is strictly related to the tolerant genotype. From this point of view, the data suggest that M4 drought tolerance could be associated to enhanced scavenging capacity of secondary ROS. In water-stressed M4 plants, the higher ROS detoxification ability could allow lateral root growth to be maintained that results in higher water uptake capacity from the soil, as observed by [START_REF] Tsukagoshi | Defective root growth triggered by oxidative stress is controlled through the expression of cell cycle-related genes[END_REF]. Likewise, at the leaf level a higher E and A n , in the drought tolerant genotype would promote active plant growth and photosynthesis. On the contrary, in 101.14 where the oxidative stress is not efficiently counteracted, the functionality of roots and leaves is strongly impaired. The genes and mechanisms found in this study to be the main factors underlying the better adaptation to water stress of the M4 genotype will be further validated using an association genetics approach. The expression of selected candidate genes is currently under evaluation on a large range of genotypes exhibiting differential responses to water stress in order to assess whether the drought tolerance strategies operating in M4 rootstock are conserved in other genotypes and, if so, to use the identified genes as functional markers for the selection of WS-tolerant grapevine rootstocks.

Figure 1 .

 1 Figure 1. Average ± SE values of net CO 2 assimilation (An) and leaf transpiration (E) for water stress plants. Results are expressed as a percentage of control. T1-4 represent sampling time points throughout the experimental period after control (T0). Values indicated with the same letters do not significantly differ according to Duncan's test (P<0.01).

Figure 2 .

 2 Figure 2. Venn diagrams of DEGs resulting from multifactorial analyses conducted on root and leaf tissues under WS treatment, according to a p<0.05. Number of DEGs influenced by each component are given in brackets; total number of genes influenced by root and leaf tissues are also given ("Total:").

Figure 3 .

 3 Figure 3. Heat map showing the expression of 101.14 and M4 genes involved in roots (A, C) and leaves (B, D) secondary metabolites biosynthesis (A, B) and transcription factors (C, D). Yellow (value = 1) and blue (value = -1) indicate up-and downregulated genes upon water stress (log 2 WS/WW mRNAseq counts), respectively. V1 12X identifier (Vitis ID) and gene name are also indicated in the figure.

ACKNOWLEDGEMENTS

This article is dedicated to the memory of our friend and colleague Angelo Ramina, we miss his constructive criticisms and suggestions. This study was supported by the Progetto AGER "SERRES", grant n° 2010-2105.

Literature cited