Planta (2014) 239:1129–1137 DOI 10.1007/s00425-014-2045-y

Emerging Technologies

Transverse mechanical properties of cell walls of single living plant cells probed by laser‑generated acoustic waves Atef Gadalla · Thomas Dehoux · Bertrand Audoin 

Received: 6 January 2014 / Accepted: 7 February 2014 / Published online: 11 March 2014 © Springer-Verlag Berlin Heidelberg 2014

Abstract  Probing the mechanical properties of plant cell wall is crucial to understand tissue dynamics. However, the exact symmetry of the mechanical properties of this anisotropic fiber-reinforced composite remains uncertain. For this reason, biologically relevant measurements of the stiffness coefficients on individual living cells are a challenge. For this purpose, we have developed the single-cell optoacoustic nanoprobe (SCOPE) technique, which uses laser-generated acoustic waves to probe the stiffness, thickness and viscosity of live single-cell subcompartments. This all-optical technique offers a sub-micrometer lateral resolution, nanometer in-depth resolution, and allows the non-contact measurement of the mechanical properties of live turgid tissues without any assumption of mechanical symmetry. SCOPE experiments reveal that single-cell wall transverse stiffness in the direction perpendicular to the epidermis layer of onion cells is close to that of cellulose. This observation demonstrates that cellulose microfibrils are the main load-bearing structure in this direction, and suggests strong bonding of microfibrils by hemicelluloses. Altogether our measurement of the viscosity at high frequencies suggests that the rheology of the wall is dominated by glass-like dynamics. From a comparison with literature, we attribute this behavior to the influence of the pectin matrix. SCOPE’s ability to unravel cell rheology Electronic supplementary material  The online version of this article (doi:10.1007/s00425-014-2045-y) contains supplementary material, which is available to authorized users. A. Gadalla · T. Dehoux (*) · B. Audoin  University Bordeaux, I2M, UMR 5295, 33400 Talence, France e-mail: [email protected]‑bordeaux1.fr A. Gadalla · T. Dehoux · B. Audoin  CNRS, I2M, UMR 5295, 33400 Talence, France

and cell anisotropy defines a new class of experiments to enlighten cell nano-mechanics.

Introduction Cell walls are a universal feature of plant cells. They control cell morphology and regulate turgor pressure to allow tissue growth (Hansen et al. 2011; Hayot et al. 2012; Routier-Kierzkowska et al. 2012), upon receiving external signaling such as hormones or environmental cues (Schopfer 2006). It is a complex composite material consisting of semi-crystalline cellulose microfibrils cross-linked by hemicelluloses, and embedded in a hydrated gel-like matrix of pectins (Hansen et al. 2011; Suslov and Verbelen 2006; Davies and Harris 2003). The mechanical properties of the wall reflect the conformation of cellulose and of pectin chains (Wilson et al. 2000), and reveal the activity of the cell itself. Indeed, pectins are notably involved in the maintaining of the fluidity within the pectin network to regulate stomatal opening in guard cells (Jones et al. 2003). It has also been demonstrated that the wall of mutants has altered mechanical properties (Hansen et al. 2011). It is, therefore, crucial to probe the mechanical properties of the wall to understand larger scale tissue dynamics. Similarly to what is observed in polymer networks (Ferry 1970), the wall composition yields a frequencydependent mechanical response to stress, appealing for dynamic probing with a broad frequency sweep. Such measurements performed on millimeter-scale tissues using extensiometry (Thompson 2001), plane–plane rheometry (Hansen et al. 2011; Whitney et al. 1999), or at subcell scale using dynamic nanoindentation (Hayot et al. 2012), have demonstrated that this frequency-dependent behavior reflects conformational changes in the wall structure, and

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that it is notably sensitive to the separation between microfibrils (Thompson 2008). These techniques function up to a few kHz, probing the fluid-like and rubber-like regions of the relaxation spectrum (Verdier et al. 2009). However, differential scanning calorimetry measurements on soybean cell walls have also revealed the existence of a glass transition (Lin et al. 1991), as observed in mammalian cells (Fabry et al. 2001; Tseng et al. 2004), believed to originate, in plant cells, from the rheological behavior of the pectin matrix (Ha et al. 1997). Since the glassy state in glass-forming polymer is best probed at GHz frequencies (Surovtsev et al. 1998; Dehoux et al. 2012), measurements in this high-frequency range should complement the existing knowledge and shed a new light on the wall rheology. Several methods to probe the mechanical properties of whole plant tissues are available, mostly inherited from material science (Cosgrove 1993). However, the intricate cellular structure forbids inferring the cell wall properties from the macroscale tissue response (Whitney et al. 1999). To circumvent this difficulty, experiments can be performed on isolated walls, or by making use of synthetic model polymer networks mimicking the wall structure (Whitney et al. 1999). Ultimately, to consider live tissues and observe cell-to-cell variations (Bryan et al. 2010), measurements at a subcell scale are necessary (Routier-Kierzkowska et al. 2012; Burgert 2006). To this end, atomic force microscopybased techniques have been employed as pressure probes (Hansen et al. 2011; Hayot et al. 2012; Burgert 2006). Yet the measurements obtained by these techniques on turgid tissues result from the complex contributions of the mechanical properties of the wall, turgor pressure and variation of the wall thickness, therefore, requiring the use of complicated modeling to infer the sole influence of the wall (Hayot et al. 2012), or the use of a varying load sufficient to stretch or rupture the wall (Routier-Kierzkowska et al. 2012). Furthermore, the microfibril orientation yields a strongly anisotropic mechanical behavior to permit in-plane elongation while resisting normal turgor pressure (Hayot et al. 2012; Routier-Kierzkowska et al. 2012; Suslov and Verbelen 2006; Reiterer et al. 1999). Extensiometry has indeed demonstrated that the preferred mean orientation of the microfibrils along the cell axis, notably in onion cells, yields a higher stiffness in this direction (Kerstens et al. 2001). Contacting-probe-based techniques such as nanoindentation (Milani et al. 2011) or cellular force microscopy (Routier-Kierzkowska et al. 2012) probe a mixture of the normal stiffness and of the in-plane stiffnesses. A precise modeling of the indentation process is then necessary to extract the different elasticity components, generally assuming an orthotropic symmetry, with a symmetry axis aligned with the microfibrils (Milani et al. 2011; Jäger et al. 2011). However, the transverse and longitudinal stiffnesses have

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never been measured independently and directly on turgid tissues, and the biological ground for this supposed symmetry remains uncertain. Indeed, the out-of-plane microfibril angle (Reiterer et al. 1999; Jäger et al. 2011), or the possible entanglement or cross-linking by hemicelluloses (Whitney et al. 1999) could break the transverse isotropy. To perform biologically relevant measurements on living, normally hydrated, turgid cells, it is therefore necessary to develop non-contact techniques, offering subcell resolution, operating on a higher frequency range, and that are able to probe the stiffness coefficients of the wall independently. In this article, we report on the use of laser-generated acoustic waves—namely the picosecond ultrasonics technique (PU)—to probe the stiffness, thickness and viscosity of live single-cell subcompartments. This non-contact technique offers a broad frequency range, extending up to 1 THz, a sub-micrometer lateral resolution and nanometer in-depth resolution (Thomsen et al. 1984, 1986; Maris 1998). Traditionally devoted to the investigation of solids, PU has allowed the mapping of mechanical, thermal and optical properties of metals, semi-conductors and amorphous materials. The use of tightly focused laser beams has given access to the mechanical anisotropy of solid structures (Audoin et al. 2008). Intricate geometries such as single microspheres (Dehoux et al. 2009; Guillet et al. 2009) or microfibers (Ségur et al. 2010) have been probed. However, despite scarce measurements of the mechanical properties of liquids (Shelton et al. 2005; Wright et al. 2008; Pezeril et al. 2009), or of single soft multiple-phase core/shell microcapsules (Dehoux et al. 2012), the use of PU to investigate soft matter remains elusive and extremely challenging. Recently, very promising applications to single-cell vacuoles have been developed (Rossignol et al. 2008). The biological demonstration of this technique is yet to clearly identify the observed structures at the probed length scale and their significance in representing physiological conditions. The detection mechanism involved in PU exploits Brillouin light scattering (BLS), an inelastic process similar to Raman scattering. Raman-based techniques rely on the scattering of light by optical phonons, and operate at THz frequencies. They provide information on the molecular structure of the sample. By contrast, Brillouin scattering uses the scattering of light by acoustic phonons (Kittel 1986). Since acoustic phonons have lower energies than optical phonons, Brillouin scattering produces frequency shifts in the GHz range that reveal the rheological behavior of the sample. Recent observation of BLS with a confocal microscope has allowed probing the viscoelastic properties of mouse cornea with a 60 μm axial resolution. However, the weakness of the scattered signal hampers lowering of the resolution further (Scarcelli and Yun 2008). To overpass this limitation, PU stimulates the scattering by the controlled emission of

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ultrashort laser-generated sound pulses. This allows observing the light scattering from nanometric objects. To illustrate the application to cellular biology of BLS enhanced by the controlled emission of acoustic phonons, we here extend PU to the remote probing of the transverse stiffness of single-cell walls of various thicknesses. We describe the related data analysis, and the subsequent theoretical interpretation. We term this approach as singlecell optoacoustic nanoprobe (SCOPE). The small amplitude of the applied deformation, together with the remote laser generation and detection, makes SCOPE inherently and completely non-invasive (Routier-Kierzkowska et al. 2012). Thus, SCOPE permits the probing of the rheological properties of individual subcell compartments with a sub-micrometer lateral resolution and a nanometer in-depth resolution. SCOPE experiments reveal that single-cell wall transverse stiffness in the direction perpendicular to the epidermis layer of onion cells is close to that of cellulose, demonstrating that microfibrils are the main load-bearing structure in this direction. Altogether our results suggest that the wall viscosity at GHz frequencies is affected by the glass-like rheology of the pectin matrix. SCOPE’s ability to unravel cell rheology and to probe directly the transverse stiffness of the wall, which is challenging in this direction without any assumption on symmetry, defines a new class of experiments to enlighten cell mechanics.

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Fig.  1  a Microphotograph of onion epidermis showing the typical individual cells. The scale bar represents 200 μm. b, c Sample geometry for the onion epidermal strip deposited on a 300 nm Ti film with b the vacuole in contact with the Ti film or c the cell wall in contact with the Ti film. The pump is focused on the bottom side of the Ti film to reduce thermal stress in the cell

Results and discussion Laser‑generated GHz acoustic waves to probe single‑cell wall dynamics Common onion (Allium Cepa L.) was used as a model system for investigating the architecture of plant cell walls (Wilson et al. 2000; Vanstreels et al. 2005). To warrant the generation of acoustic waves through the photoelastic interaction, and to avoid corrosion and ion release, strips of onion cells were placed onto a biocompatible transducer, composed of a 300 nm layer of titanium (Ti) sputtered onto a transparent SiO2 substrate, as detailed by Dehoux and Audoin (2012). A top-view white-light image of the cells is shown in Fig. 1a. The side of strip oriented towards the bulb core can be placed facing upwards, as in our previous works (Rossignol et al. 2008; Dehoux and Audoin 2012). In this configuration, the thinnest part of the wall is in contact with the transducer. As the thickness of this wall region is much smaller than the acoustic wavelength ~280 nm, only the cell vacuole is probed. For this reason, in the remainder we consider the vacuole to be in acoustic contact with the transducer in such configuration. In the present study, we now also alternatively place the strip facing downwards so that the thickest part of the wall is in contact with the transducer. This second

Fig. 2  Simplified scheme of the setup. To reduce temperature rise at cell-transducer interface, the pump and probe beams are focused on the bottom and on the top surfaces of the Ti film, respectively

configuration allows direct probing of the mechanical properties of the wall. The geometry of sample is represented schematically in both configurations in Fig. 1b, c. We use an optical pump-probe technique described schematically in Fig. 2. Ultrashort optical pulses of duration 100 fs and wavelength λ = 800 nm are generated by a Ti:Sapphire laser. Part of the laser beam passes through a doubling crystal to change the wavelength of the pump pulse trains to 400 nm. To increase the sensitivity to optical reflectivity variations, the pump trains are chopped at a frequency fm for lock-in detection. The 800 nm probe beam passes through a mechanical delay line to tune the pumpprobe time delay t. Collinear pump and probe beams are focused on the bottom and on the top sides of the Ti film, respectively. In such configuration, we have demonstrated in a previous article (Dehoux and Audoin 2012) that neither the pump radiation, the transient-temperature rise, nor

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the overheated electrons reach the top Ti surface. This is an important advantage when probing biological material in physiological conditions. The modulation frequency fm and the Ti film thickness d are chosen so that the thermal diffusion length remains smaller than d. Thereby no thermal confinement occurs in the Ti film and the stress imposed to the cell is reduced importantly. These precise adjustments also lead to a significant increase in the signal-to-noise ratio. The pump light is absorbed in Ti in the vicinity of the Ti/SiO2 interface over a nanometer depth comparable with the optical skin depth. The subsequent ultrafast thermal expansion launches a longitudinal acoustic pulse in Ti, with a broad spectrum extending up to 200 GHz. The absorbed heat simultaneously diffuses on a larger timescale, and partially sinks into the SiO2 substrate. The acoustic pulse propagates through the Ti film and, owing to the cell-transducer intimate contact, is transmitted to the cell. The acoustic strain pulse then propagates in the cell subcompartments. We illustrate this phenomenon with an animated scheme (see supplementary movie 1 online). The acoustic propagation is measured with laser probe pulses. The acoustic phonons scatter the probe light and the subsequent transient reflectivity change δR is recorded via a photodiode as a function of the pump-probe time delay. Typical signal obtained on the bare transducer is shown in Fig. 3a. Acoustic echoes arising from successive reflections within the Ti film are indicated with downward arrows. The Fourier spectrum of a single echo plotted in Fig. 3b demonstrates the wide frequency bandwidth extending up to 200 GHz obtained with such a transducer, with a maximum located around 50 GHz. This all-optical technique, therefore, offers a non-contact mean to probe single-cell subcompartments with a diffraction-limited submicrometric lateral resolution. The broad GHz bandwidth offers an innovative mean to investigate the high-frequency rheological behavior of biopolymers such as cellulose.

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Fig.  3  a Optical reflectivity variation δR for the bare Ti/SiO2 transducer as a function of pump-probe time delay. Acoustic echoes arising from successive reflections within the Ti film are indicated with downward arrows. b Amplitude of the Fourier spectrum of the first echo

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Measurements in single‑cell vacuoles The optical reflectivity variation δR measured when the vacuole is against the transducer top surface is plotted in Fig. 4a. In addition to the acoustic echoes previously measured in Fig. 3a for the bare transducer, indicated with downward arrows in Figs. 3a and 4a, a damped oscillation is now observed. Since the cell is transparent at the probe wavelength λ = 800 nm, the probe light is partially reflected at the cell-transducer interface and, owing to the photoelastic effect, partially scattered by the strain pulse propagating inside the cell. We illustrate this phenomenon with an animated scheme (see supplementary movie 2 online). These two optical contributions

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Fig. 4  Optical reflectivity δR measured (plain line) when a the vacuole is against the transducer, b a thick wall is against the transducer and c a thin wall is against the transducer. Theoretical calculations are plotted with dotted lines. The first two echoes are indicated with downward arrows

interfere and give rise to oscillations of optical reflectivity as the acoustic wave propagates. Such oscillations are called Brillouin oscillations (Thomsen et al. 1986). Brillouin oscillations only occur when the optical probe and acoustic wavevectors are matched. Their frequency is therefore

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Fig. 5  Longitudinal sound velocity in the out-of-plane direction for the cell vacuole (left n = 13) and cell wall (right n = 8)

fi =

2ni Vi , 

(1)

where ni and Vi are the refractive index and longitudinal sound velocity, respectively (index i = v, w stands for the vacuole and the wall). Equation 1 shows that the frequency fi of the Brillouin oscillations is proportional to the sound velocity Vi. Since the refractive index of Allium Cepa cells is well documented (Liu et al. 2008), we can determine Vi directly for a given laser wavelength λ. The lifetime of the oscillations τi, related to the relaxation processes, provides a measure of the sound attenuation Γi = 1/Viτi. The inverse of sound attenuation, 1/Γi, is the distance over which the amplitude of the wave has decreased by 1/e. A fit with a function sin(2πfv)exp(−VvΓvt), plotted with a red dashed line in Fig. 4a, gives a frequency fv =  5.4 GHz and a lifetime 1/τv  = 2.6 ns−1 for the vacuole averaged over 13 cells. Given that nv = 1.35 (Liu et al. 2008), the sound velocity is Vv = 1,600 m s−1 and the sound attenuation is Γv = 1.6 μm−1. The distribution of sound velocity values is plotted in Fig. 5. These values are in excellent agreement with those found in the literature (Audoin et al. 2010). Measurements in single‑cell walls The thickness of the wall can vary depending on the cell and on the position in the cell (Hayot et al. 2012). Two typical optical reflectivity variations δR measured when the wall is against the transducer top surface are plotted in Fig. 4b, c, corresponding to different thicknesses. The reflectivity plotted in Fig. 4b is similar to that observed in the vacuole because the wall thickness d is greater than the attenuation length Γ−1 w and the sound pulse is attenuated before reaching the wall–vacuole interface. In this case, the velocity Vw and attenuation Γw can be obtained as presented above, using a fit with a damped sine function, as

plotted with a dashed line in Fig. 4b. The reflectivity measured for cells with a thinner wall is plotted in Fig. 4c. This time the oscillations stop after ~350 ps upon reaching the wall–vacuole interface. We observe a step-like reflectivity variation due to acoustic-induced motion of the wall– vacuole interface. The step-like variation is of relatively smaller amplitude than that observed in mammalian cells (Ducousso et al. 2011) owing to the low optical mismatch across the wall–vacuole interface (Liu et al. 2008). To illustrate the physics of signal formation in thin regions of the wall, we model the photoelastic interaction as detailed by Ducousso et al. (2011). The wall structure is described as a finite layer, sandwiched between semi-infinite Ti and vacuole layers. Heat diffusion following laser absorption in Ti is described by the Fourier equation using the thermal properties of pure Ti (Audoin et al. 2010). Sound propagation resulting from the temperature rise is described by a 1D wave equation. A damping of the form exp(−Γwx) accounts for sound attenuation in the wall. The optical reflectivity change δR is obtained by solving Maxwell’s equation written for a dielectric permittivity modulated by sound propagation (Thomsen et al. 1986). The calculation is fitted to the data by adjusting the longitudinal sound velocity Vw and wall thickness d to match the oscillations and step-like reflectivity change, respectively. Calculated δR is plotted with dashed lines in Fig. 4c. From this comparison, we find that the wall thickness can be as small as 800 nm (see Fig. 4c) and that, in the case where the sound pulse is attenuated before reaching the wall–vacuole interface, it can also be greater than 1.6 μm (see Fig. 4b). Such values are in good agreement with those determined by nanoindentation (Hayot et al. 2012) or optical microscopy (Vanstreels et al. 2005). The average Brillouin frequency is fw = 12 GHz. Note that fw could also be obtained directly using a Fourier transform for instance, without the need of the illustrative model presented above. Given the refractive index in the wall nw = 1.41 (Liu et al. 2008), the sound velocity is Vw = 3,410 m s−1. The distributions of measured sound velocities are plotted in Fig. 5. Let us examine these results in more details with some simple considerations from the homogenization theory.

Discussion Role of the xyloglucans in the out‑of‑plane longitudinal stiffness of the wall The wall is mainly composed of cellulose microfibrils cross-linked by hemicelluloses, and embedded in a hydrated gel-like matrix of pectins (Hansen et al. 2011; Suslov and Verbelen 2006; Davies and Harris 2003). The microfibrils are oriented along the cell axis (Kerstens et al.

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2001), yielding a strongly anisotropic mechanical behavior. The fibrils are widely believed to be connected by hemicelluloses, in particular xyloglucans, forming bridges held together by hydrogen bonds (Burgert and Fratzl 2007). The acoustic wavelength is here of λ/2nw = 280 nm in the wall, and the largest structures in the probed area are crystalline portions of the microfibrils, > Mh′ Eq. 3 implies that Mw′ should tend towards Mh′ /f . This is however much lower

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than what we measure, meaning that the mechanical properties of the wall are instead dominated by the cellulose content, as already observed for the shear modulus (Whitney et al. 1999). In this light, it has been suggested that the cell wall rather behaves like an isostrain system (Thompson 2005). This has been explained by role of xyloglucans acting as a spacer between cellulose microfibrils. We here provide evidence of a similar behavior for the out-of-plane longitudinal stiffness. Glass‑like behavior of the pectin matrix The lifetime of the Brillouin oscillations in the wall τw is related to the relaxation mechanisms. We determine the sound attenuation 1/τw = 4.6 ns−1 (n = 4) for cells where an exponential decay can be recognized. This value converts to the sound attenuation Γw  = 1/τwVw  = 1.3 μm−1. Using the following formula for the longitudinal loss modulus (Litovitz and Davis 1965), ′′

Mw =

2Vw3 Γw ρw , 2πfw

(4)

we obtain M″w= 1.7 GPa. This quantity reflects the mechanical energy dissipated as heat by viscous processes. It is also interesting to evaluate the ratio of viscous losses to elastic storage δ = Mw′′ /Mw′ = 0.13, and note that this is a typical value for a glass-forming polymer in the glassy state. Indeed, dynamic scanning calorimetry measurements have revealed a glass transition, attributed to the pectin matrix (Lin et al. 1991). We thus propose that, in contrast with the storage modulus Mw′ , the loss modulus Mw′′ reflects both the dynamic behavior of the cellulose fibrils and pectin matrix in the glassy state. However, to separate the contributions of the fibrils and matrix, more advanced homogenization theories should be developed (Waterman 1969), requiring independent knowledge of the rheological behavior of each of these components. Measurements on isolated components should help performing such a detailed analysis.

Conclusion We have shown that SCOPE can be used to probe the rheological behavior of single cells with a lateral resolution limited by optical diffraction only. Such measurements give access to the GHz region of the relaxation spectrum, and complement the range accessible by other techniques. Since this technique is non-contact, it can be used on live turgid tissues. Future development of the detection of lasergenerated acoustic waves at various probe wavelengths will permit the exploration of a wide frequency range, shedding new light on the rheological behavior of cells. In addition,

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fluorescent markers can be used simultaneously. Subsequent integration of this technique with state-of-the-art cell micromanipulation tools and imaging platforms will open new areas for the investigation of physiological processes at a subcell scale.

Methods Acousto‑optical setup The main laser beam is split into two beams, the pump and probe beams, using a polarizing beam splitter. The pump beam is frequency-doubled by a non-linear Beta Barium Borate (BBO) crystal to obtain a wavelength of 400 nm. It is then optically chopped at a frequency fm for lock-in detection by an acousto-optic modulator. The modulation frequency fm  = 1 MHz and the Ti film thickness d = 300 nm are chosen so that the thermal diffusion length,  κ xd = , (5) Cp πfm

at fm, where κ = 3.1 W m−1 K−1 and Cp = 500 J kg−1 K−1 are the thermal conductivity and heat capacity of Ti, remains smaller than d. Collinear pump and probe beams are focused on the bottom and on the top sides of the Ti film using a 60× (NA 0.7) and a 50× objective lens (NA 0.8), respectively. The 1/e2 radius of the probe spot size is measured from the Gaussian intensity profiles recorded by a CCD camera at 1 μm at the focal point on a bare metallic sample (Higuet et al. 2011). Typical laser energies are 9 pJ for the pump and 1.5 pJ for the probe. The pump pulses are initially absorbed at the Ti/SiO2 interface in Ti over a distance comparable to the optical skin depth, approximately 15 nm (Audoin et al. 2010). This distance is much smaller than the Ti film thickness d, and no overheated electrons reach the top Ti surface. The room temperature is maintained at 20 °C during the whole experiment for laser stability.

Typical values of n″t yield γ−1 ≈ 15 nm at the pump wavelength (Audoin et al. 2010). Ras and Rst are the coefficients of optical reflection in intensity at the air-SiO2 and SiO2titanium interfaces, respectively. We model the photoelastic interaction as detailed by Ducousso et al. (2011). We first describe the acoustic generation. The radius of the Gaussian probe beam is by one order of magnitude larger than the wavelength of the acoustic phonons and no acoustic diffraction occurs. The cell is thus described as a finite layer lying on a semi-infinite Ti6Al4V half-space. Temperature rise Tt(x,t) in Ti following laser absorption in Ti6Al4V is described by the Fourier equation using the thermal properties of pure Ti (Audoin et al. 2010). Equation 4 is the source term of the Fourier heat equation in titanium. We consider that the thermal properties of the wall, taken equal to those of water, have negligible influence since the velocity of thermal expansion at the Brillouin frequencies is slower than the acoustic velocity (Dehoux et al. 2008). The temperature rise Ti is thereby obtained in each layer of index i. The ensuing mechanical expansion αi∇Ti, where αi is the thermal dilatation coefficient of medium i, is introduced into the elastic wave equation as a source term. The thermal equation and wave equation thus form a coupled system of differential equations that is solved analytically in the Fourier space. The propagation of the longitudinal strain ηi(x,ω) in the wall is thus obtained, with ω the Fourier frequency. A damping of the form exp(−Γix) accounts for phonon attenuation Γi in the cell. The detection mechanism is modeled as follows. The strain ηi(x,ω) modulates the dielectric permittivity of the cell of a quantity δε  ∝  ηi(x,ω). The optical reflectivity ˜ change δ R(ω) is obtained in the Fourier space by solving Maxwell’s equation written for a dielectric permittivity δε(x,ω) modulated by sound propagation. The time-domain solution δR(t) is then obtained using an inverse Fourier transform (Dehoux et al. 2007) computed using the MATLAB software. Sample preparation

Photoelastic modeling The pump laser beam propagates at normal incidence in the direction x. The absorbed electromagnetic energy is expressed from the Poynting vector (Landau and Lifchitz 1969) in the titanium layer,

Q(x, t) = γ I0 (1 − Ras Rst )f (t)e−γ x ,

(6)

with I0 the incident energy per unit length. Function f(t) is the time Gaussian distribution of intensity of the modelocked laser pulses. γ−1 = λ/4πn″t is the optical penetration depth in titanium, where λ and n″t are the laser wavelength and the imaginary part of the refractive index in titanium.

Common onion (Allium Cepa L.) was acquired commercially for all experiments. The second scale of fresh and healthy onion bulbs was selected to avoid either not fully hydrated cells from the outermost layer or non-mature cells from the inner bulb core. The epidermis was cut longitudinally, parallel to the vascular bundles, into rectangular pieces with a clean surgical knife. These epidermal strips with approximate dimensions 2 × 6 mm were carefully removed from the equatorial region of the inner tissue surface. A top-view white-light image of the cells is shown in Fig. 1c. We checked by optical microscopy that a single layer of cells was obtained. Glass cover slips were sputtered

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with Ti by RF sputtering to obtain a thickness of 300 nm estimated from the sputtering time. The onion strips were placed on the Ti top side and gently pressed. All measurements were performed in the flat portion of the cell, far away from the mid lamella located between adjacent cells. Determination of the sound velocity In all cases, the frequency fi of the Brillouin oscillations observed in medium i is measured directly without any modeling. The precision on the sound velocity is given by:

∆ni ∆fi ∆vi = + . vi ni fi

(7)

It has been observed that the optical index ni varies little, even when comparing the wall and vacuole (Liu et al. 2008). We, therefore, estimate the maximum variation from cell to cell to be much smaller than the variation from the wall to the vacuole: Δni/ni  ≪ (nw  −  nv)/ni  ≪ 0.05. The observed precision in the frequency fi is Δfi/fi 

Transverse mechanical properties of cell walls of single living plant cells probed by laser-generated acoustic waves.

Probing the mechanical properties of plant cell wall is crucial to understand tissue dynamics. However, the exact symmetry of the mechanical propertie...
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