Nicolas Vuillerme
email: nicolas.vuillerme@imag.fr

Matthieu Boisgontier

Olivier Chenu

Jacques Demongeot

Yohan Payan

TONGUE-PLACED TACTILE BIOFEEDBACK SUPPRESSES THE DELETERIOUS EFFECTS OF MUSCLE FATIGUE ON JOINT POSITION SENSE AT THE ANKLE

Keywords: Sensory re-weighting, Biofeedback, Proprioception, Muscle fatigue, Tongue Display Unit, Ankle

Introduction

Proprioceptive afferent from the ankle joint is known to play a significant role in the control of human posture and gait that greatly influence our ability to perform activities of daily living. Impaired ankle proprioception also may be a predisposing factor for chronic ankle instability, balance difficulties, reduced mobility functions, fall, injury and re-injury (e.g. Fu Interestingly, recent studies reporting deleterious effects of localized fatigue of the ankle muscles on joint position sense [START_REF] Forestier | Alteration of the position sense at the ankle induced by muscular fatigue in humans[END_REF]) and balance control (e.g. Ledin et al. 2004; Vuillerme et al. 2006a; Vuillerme and Demetz 2007; Vuillerme et al. 2002aVuillerme et al. ,b, 2003) ) support this view. Accordingly, there is a need to develop techniques for enhancing proprioceptive acuity at the ankle. A promising intervention, using an artificial tongue-placed tactile biofeedback, has recently been shown to improve ankle joint position sense in young healthy adults (Vuillerme et al. 2006b,c). The underlying principle of this system consisted of supplying individuals with supplementary information about the position of their matching ankle position relative to their reference ankle position through a tongue-placed tactile output device generating electrotactile stimulation of the tongue (Tongue Display Unit, TDU) (Bachy-Rita et al. 1998; Bach-y-Rita and Kercel 2003) (Figure 1). Note that the tongue was chosen as a substrate for electrotactile stimulation site according to its neurophysiologic characteristics. Indeed, because of its dense mechanoreceptive innervations [START_REF] Trulsson | Low-threshold mechanoreceptive afferents in the human lingual nerve[END_REF] and large somatosensory cortical representation [START_REF] Picard | Sensory cortical tongue representation in man[END_REF]), the tongue can convey higher-resolution information than the skin can [START_REF] Sampaio | Brain plasticity: 'visual' acuity of blind persons via the tongue[END_REF][START_REF] Van Boven | The limit of tactile spatial resolution in humans: grating orientation discrimination at the lips, tongue, and finger[END_REF]. In addition, due to the excellent conductivity offered by the saliva, electrotactile stimulation of the tongue can be applied with much lower voltage and current than is required for the skin (Bach-y-Rita et al. 1998). At this point however, although the above-mentioned studies evidenced that this artificial tongue-placed tactile biofeedback can be used to improve ankle joint proprioception under normal ankle neuromuscular state (i.e.

with redundant and reliable sensory information) (Vuillerme et al. 2006b,c), its effectiveness under altered ankle proprioceptive conditions, as it is the case following muscle fatigue [START_REF] Forestier | Alteration of the position sense at the ankle induced by muscular fatigue in humans[END_REF], has not been established yet. Within this context, the purpose of the present experiment was to investigate whether this biofeedback could mitigate the deleterious effect of muscle fatigue on joint position sense at the ankle. It was hypothesized that (1) ankle muscles fatigue would decrease ankle joint position sense without the provision of the biofeedback delivered through the TDU, (2) biofeedback would improve ankle joint position sense in the absence of ankle muscles fatigue, and, more originally (3) biofeedback would mitigate the detrimental effect of ankle muscles fatigue on ankle joint position sense.

Methods

Subjects

Sixteen male young healthy university students (age = 22.3 ± 2.4 years; body weight = 72.1 ± 9.5 kg; height = 180.8 ± 6.2 cm) voluntarily participated in the experiment. Criteria for selection and inclusion were: male; aged 20-30 years; negative medical history; normal ankle range of motion. Exclusion criteria were: history of sprain, fracture, injury, surgery of the lower extremities; restricted ankle range of motion of less than 20° of dorsiflexion or 30° of plantarflexion; history of central or peripheral nervous system dysfunctions. Subjects were familiarized with the experimental procedure and apparatus, and they gave their informed consent to the experimental procedure as required by the Helsinki declaration (1964) and the local Ethics Committee. None of the subjects presented any history of injury, surgery or pathology to either lower extremity that could affect their ability to perform the ankle joint position sense test.

Apparatus for measuring ankle joint position sense

Ankle joint position sense measurements were carried out as previously done by Vuillerme et al. (2006b,c). Subjects were seated barefoot in a chair with their right and left foot secured to two rotating footplates with Velcro straps. The knee joints were flexed at about 110°. Movement was restricted to the ankle in the sagittal plane, with no movement occurring at the hip or the knee. The axes of rotation of the footplates were aligned with the axes of rotation of the ankles. Precision linear potentiometers attached on both footplates provided analog voltage signals proportional to the ankles' angles. A press-button was held in the right hand and served to record the matching. Signals from the potentiometers and the press-button were sampled at 100 Hz (12 bit A/D conversion), then processed and stored within the Labview 5.1 data acquisition system. In addition, a panel was placed above the subject's legs to eliminate visual feedback about both ankles position.

Apparatus for providing tactile biofeedback

The underlying principle of the biofeedback device, similar to that recently used by ----------------------------------Insert Figure 1 about here ----------------------------------The following coding scheme for the TDU was used (Vuillerme et al. 2006c): (1) no electrical stimulation when both ankles were in a similar angular position within a range of 0.5°; (2) stimulation of either the anterior or posterior zone of the matrix (2 × 6 electrodes) (i.e. stimulation of front and rear portions of the tongue) depending on whether the matching right ankle was in a too plantarflexed or dorsiflexed position relative to the reference left ankle, respectively. The frequency of the stimulation was maintained constant at 50 Hz across subjects, ensuring the sensation of a continuous stimulation over the tongue surface.

Conversely, the intensity of the electrical stimulating current was adjusted for each subject, and for each of the front and rear portions of the tongue, given that the sensitivity to the electrotactile stimulation was reported to vary between individuals [START_REF] Essick | Lingual tactile acuity, taste perception, and the density and diameter of fungiform papillae in female subjects[END_REF]), but also as a function of location on the tongue in a preliminary experiment (Vuillerme et al. 2006b).

Task and procedure

The experimenter first placed the left reference ankle at a predetermined angle where the position of the foot was placed on a wooden support. By doing so, subjects did not exert any effort to maintain the position of the left reference ankle, preventing the contribution of effort cues coming from the reference ankle to the sense of position during the test (e.g., Vuillerme et al. 2006b,c;[START_REF] Walsh | Human forearm position sense after fatigue elbow flexor muscle[END_REF][START_REF] Winter | Muscle spindle signals combine with the sense of effort to indicate limb position[END_REF]. A matching angular target position of 10° of plantarflexion was selected to avoid the extremes of the ankle range of motion to minimize additional sensory input from joint and cutaneous receptors (e.g., Burke et al. 1988). Once the left foot had been positioned at this test angle, subjects' task was to match its position by voluntary placement of their right ankle. When they felt that they had reached the target angular position (i.e., when the right foot was presumably aligned with the left foot), they were asked to press the button held in their right hand, thereby registering the matched position. Subjects were not given feedback about their performance and were not given any speed constraints other than 10-s delay to perform one trial. Results of a preliminary study using 10 subjects established the "excellent" test-retest reliability of the active ankle-matching task (intra-class correlation coefficient (ICC) > 0.75).

Two sensory conditions were presented, (1) the No-biofeedback serving as a control condition and (2) the Biofeedback condition in which the subjects executed the active matching task using a TDU-biofeedback system, as described above. These two sensory conditions also were executed under two states of ankle muscles fatigue. [START_REF] Allen | Effect of muscle fatigue on the sense of limb position and movement[END_REF] The No-fatigue condition served as a control condition, whereas (2) in the Fatigue condition, the measurements were performed immediately after a fatiguing procedure. Its aim was to induce a muscular fatigue in the ankle plantar-flexor of the right leg until maximal exhaustion.

Subjects were asked to perform toe-lifts with their right leg as many times as possible following the beat of a metronome (40 beats/min). Verbal encouragement was given to ensure that subjects worked maximally. The fatigue level was reached when subjects were no more able to complete the exercise. Immediately on the cessation of exercise, the subjective exertion level was assessed through the Borg CR-10 scale [START_REF] Borg | Psychological scaling with applications in physical work and the perception of exertion[END_REF]). Subjects rated their perceived fatigue in the ankle muscles as almost "extremely strong" (mean Borg ratings of 9.1 ± 0.6). To ensure that ankle joint proprioception measurements in the Fatigue condition were obtained in a real fatigued state, various rules were respected, as described in previous studies

Data analysis

Two dependent variables were used to assess matching performances [START_REF] Schmidt | Motor Control and Learning, 2nd Edition[END_REF]).

(1) The absolute error (AE), the absolute value of the difference between the position of the right matching ankle and the position of the left reference ankle, is a measure of the overall accuracy of positioning.

(2) The variable error (VE), the variance around the mean constant error score, is a measure of the variability of the positioning. Decreased AE and VE scores indicate increased accuracy and consistency of the positioning, respectively [START_REF] Schmidt | Motor Control and Learning, 2nd Edition[END_REF].

Statistical analysis

Data obtained for AE and VE were submitted to separate 2 Biofeedback (Nobiofeedback vs. Biofeedback) × 2 Fatigues (No-fatigue vs. Fatigue) analyses of variances (ANOVAs) with repeated measures of both factors. Post hoc analyses (Newman-Keuls) were performed whenever necessary. Level of significance was set at 0.05.

Results

Positioning accuracy

Analysis of the AE showed a significant interaction of Biofeedback × Fatigue (F(1,15) = 4.72, P < 0.05, Fig. 2A). The decomposition of the interaction into its simple main effects showed that the availability of the biofeedback suppressed the effect of Fatigue (Fig. 2A). The ANOVA also showed main effects of Biofeedback (F(1,15) = 32.65, P < 0.001) and Fatigue (F(1,15) = 23.89, P < 0.001), yielding smaller AE in the Biofeedback than No-biofeedback condition and larger AE in the Fatigue than No-fatigue condition, respectively.

Positioning variability

Analysis of the VE showed a significant interaction of Biofeedback × Fatigue (F(1,15) = 6.41, P < 0.05, Fig. 2B). The decomposition of the interaction into its simple main effects showed that the availability of the biofeedback suppressed the effect of Fatigue (Fig. 2B). The ANOVA also showed main effects of Biofeedback (F(1,15) = 12.70, P < 0.01) and Fatigue (F(1,15) = 12.60, P < 0.01), yielding smaller VE in the Biofeedback than No-biofeedback condition and larger VE in the Fatigue than No-fatigue condition, respectively.

- ---------------------------------Insert Figure 2 about here ----------------------------------

Discussion

Whereas the acuity of the position sense at the ankle can be disturbed by muscle fatigue [START_REF] Forestier | Alteration of the position sense at the ankle induced by muscular fatigue in humans[END_REF], it recently also has been shown to be improved, under normal ankle neuromuscular state, through the use of an artificial tongue-placed tactile biofeedback (Vuillerme et al. 2006b,c). The underlying principle of this biofeedback consisted of supplying individuals with supplementary information about the position of their matching ankle position relative to their reference ankle position through electrotactile stimulation of the tongue. Within this context, the purpose of the present experiment was to investigate whether this biofeedback could mitigate the deleterious effect of muscle fatigue on joint position sense at the ankle.

To address this objective, sixteen young healthy university students were asked to perform an active ankle-matching task in two conditions of No-fatigue and Fatigue of the ankle muscles and two conditions of No-biofeedback and Biofeedback. Measures of the overall accuracy and the variability of the positioning were determined using the absolute error and the variable error, respectively [START_REF] Schmidt | Motor Control and Learning, 2nd Edition[END_REF]).

On the one hand, without the provision of biofeedback (No-biofeedback condition), fatigue yielded less accurate and less consistent matching performances, as indicated by increased absolute (Fig. 2A) and variable errors (Fig. 2B) observed in the Fatigue relative to the No-fatigue condition, respectively. This result, in line with that of [START_REF] Forestier | Alteration of the position sense at the ankle induced by muscular fatigue in humans[END_REF], was expected (hypothesis 1) and also is consistent with the existing literature reporting impaired proprioceptive acuity following muscle fatigue induced at the knee (Lattanzio et al. On the other hand, in the absence of fatigue at the ankle joint (No-fatigue condition), more accurate and more consistent matching performances were observed with than without biofeedback, as indicated by decreased absolute (Fig. 2A) and variable errors (Fig. 2B), respectively. This result also was expected (hypothesis 2), corroborating those of Vuillerme et al. (2006b,c). It confirms that, under "normal" proprioceptive conditions, the central nervous system is able to integrate an artificial biofeedback signal delivered through electrotactile stimulation of the tongue to improve joint position sense at the ankle.

Finally, and more originally, the availability of the biofeedback allowed the subjects to suppress the deleterious effects of muscle fatigue on ankle positioning accuracy and variability, as indicated by the significant interactions Biofeedback × Fatigue observed for absolute (Fig. 2A) and variable errors (Fig. 2B), respectively. Confirming our hypothesis 3, these results suggest an increased reliance on biofeedback delivered through the TDU for the active ankle-matching task in condition of muscle fatigue at the ankle. We could interpret these findings relative to the concept of "sensory re-weighting" (e.g. Horak and Macpherson Oie et al. 2002;Vuillerme et al., 2002aVuillerme et al., ,2005a). Indeed, it has been proposed that, as a sensory input becomes lost or disrupted (as it was the case for the proprioceptive signals from the ankle in the Fatigue condition), the relative contributions of alternative available sensory inputs are adaptively re-weighted by increasing reliance on sensory modalities providing accurate and reliable information (as it was the case when the tactile biofeedback delivered through the TDU was made available in the Biofeedback condition). Interestingly, although the experimental task was different, this interpretation is reminiscent with previous results evidencing the adaptive capabilities of the central nervous system to cope with ankle muscle fatigue for controlling posture during bipedal quiet standing. It was observed that, to some

  et al. 2005; Halasi et al. 2005; Madhavan and Shields 2005; Payne et al. 1997).

  Vuillerme et al. (2006b,c), consisted of supplying individuals with supplementary information about the position of the matching right ankle relative to the reference left ankle position through a tongue-placed tactile output device. Electrotactile stimuli were delivered to the front part of the tongue dorsum via flexible electrode arrays placed in the mouth, with connection to the stimulator apparatus via a flat cable passing out of the mouth. The system comprised a 2D electrode array (1.5 × 1.5 cm) consisting of 36 gold-plated contacts each with a 1.4 mm diameter, arranged in a 6 × 6 matrix (Vuillerme et al. 2006b,c, 2007a,b,c) (Figure 1).

  investigating the effect of ankle muscles fatigue on postural control (Ledin et al. 2004; Vuillerme et al. 2006a; Vuillerme and Demetz 2007; Vuillerme et al. 2003). (1) The fatiguing exercise took place beside the experimental set-up to minimise the time between the exerciseinduced fatiguing activity and the proprioceptive measurements, (2) the duration of the postfatigue conditions was short (1 minute) and (3) the fatiguing exercise was repeated prior to each No-biofeedback and Biofeedback condition.

For

  each condition of No-biofeedback and Biofeedback and each condition of Nofatigue and Fatigue, subjects performed five trials, for a total of 20 trials. The order of presentation of the two No-biofeedback and Biofeedback conditions was randomised over subjects.

  1997; Skinner et al. 1986), hip (Taimela et al. 1999), neck (Vuillerme et al. 2005b), shoulder (Björklund et al. 2000; Lee et al. 2003; Voight et al. 1996), or elbow (Allen and Proske 2005; Brockett et al. 1997; Walsh et al. 2004). At this point, although the exact mechanism inducing proprioceptive impairment subsequent to the fatiguing exercise is rather difficult to answer and was not within the scope of our study, it is likely that the above-mentioned adverse effects of muscle fatigue on joint position sense stem from alterations of signals from muscle spindles (Björklund et al. 2000; Brockett et al. 1997; Forestier et al. 2002; Vuillerme et al. 2005b) and/or sense of effort (Allen and Proske 2006; Walsh et al. 2004) associated with muscle fatigue.

  extent, vision (Ledin et al. 2004; Vuillerme et al. 2006a), ankle foot orthoses (Vuillerme and Demetz 2007) and a light finger touch (Vuillerme and Nougier 2003) could compensate for the postural perturbation induced by muscular fatigue applied to the ankle, hence reflecting a re-weighting of sensory cues in balance control following ankle muscles fatigue by increasing the reliance on visual information, cutaneous inputs from the foot and shank and haptic cues from the finger, respectively. From a fundamental perspective in neuroscience, the abovementioned results and ours collectively argue in favour of a re-weighting of available, accurate and reliable sensory cues as a function of the neuromuscular constraints acting on the individual. At this point, however, while the present experiment was specifically designed to assess the combined effects of muscle fatigue and tongue-placed tactile biofeedback on joint position sense at the ankle, it could also be interesting to investigate these combined effects on the control of ankle movements in more functional tasks such as postural control during quiet standing or gait initiation. Finally, the present findings could also have implications in clinical conditions and rehabilitation practice. Biofeedback delivered through TDU may indeed enable individuals to overcome ankle proprioceptive impairment caused by trauma (Bressel et al. 2004; Halasi et al. 2005; Payne et al. 1997), normal aging (Madhavan and Shields 2005; Verschueren et al. 2002), or disease (Simoneau et al. 1996; van den Bosch et al. 1995; van Deursen et al. 1998). An investigation involving older healthy adults is being conducted to strengthen the potential clinical value of the approach reported in the present experiment.

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. Mean and standard deviation for the absolute error (A) and the variable error (B) for

Figure 2

 2 Figure 2.

Acknowledgements

The authors are indebted to Professor Paul Bach-y-Rita for introducing us to the Tongue Display Unit and for discussions about sensory substitution. Paul has been for us more than a partner or a supervisor: he was a master inspiring numerous new fields of research in many domains of neuroscience, biomedical engineering and physical rehabilitation. The authors would like to thank subject volunteers. Special thanks also are extended to Damien Flammarion and Sylvain Maubleu for technical assistance, Benjamin Bouvier for his help in data collection, Dr. Vince and Zora B. for various contributions. This research was supported by the Fondation Garches and the company IDS.


Elise Azar

Doru Constantin

Dror E Warschawski

The effect of gramicidin inclusions on the local order of membrane components

Keywords: gramicidin, membranes, NMR, WAXS, order parameter

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

The eect on the cell membrane of inclusions (membrane proteins, antimicrobial peptides etc.) is a highly active eld of study in biophysics [1]. A very powerful principle employed in describing the interaction between proteins and membranes is that of hydrophobic matching [2,3]. It states that proteins with a given hydrophobic length insert preferentially into membranes with a similar hydrophobic thickness [4].

Many studies of the interaction used as inclusion the antimicrobial peptide (AMP) Gramicidin A (GramA), which is known [5,6] to deform (stretch or compress) host membranes to bring them closer to its own hydrophobic length, so the hydrophobic matching mechanism is likely relevant. This perturbation of the membrane prole induces between the GramA pores in bilayers with various compositions [7] a repulsive interaction that can be explained based on a complete elastic model [8].

This large-scale description raises however fundamental questions about the microscopic eect of the inclusion, at the scale of the lipid or surfactant molecules composing the membrane. To what extent is their local arrangement perturbed by the inclusion? Is the continuous elastic model employed for bare membranes still valid?

In this paper, our goal is to investigate the inuence of GramA inclusions on the local order of the lipid or surfactant chains. We combine two complementary techniques: wide-angle X-ray scattering (WAXS) gives access to the positional order between neighboring chains, while nuclear magnetic resonance (NMR) is sensitive to the orientational order of chain segments, thus yielding a comprehensive picture of the state of the membrane as a function of the concentration of inclusions.

We study GramA inserted within bilayers composed of lipids with phosphocholine heads and saturated lipid chains: 1,2-dilauroyl-sn -glycero-3-phosphocholine (DLPC) and 1,2-dimyristoyl-sn -glycero-3-phosphocholine (DMPC) or of single-chain surfactants with zwitterionic or nonionic head groups: dodecyl dimethyl amine oxide (DDAO) and tetraethyleneglycol monododecyl ether (C 12 EO 4 ), respectively. the hydrophobic length of DLPC (20.8 Å) [5] DDAO (18.4 Å) [6] and C 12 EO 4 (18.8 Å) [7] is shorter than that of GramA (22 Å) [9], while DMPC (25.3 Å) [5] is longer. Since all these molecules form bilayers, and their hydrophobic length is close to that of GramA, the latter is expected to adopt the native helical dimer conguration described by Ketchem et al. [10],

and not the intertwined double helices observed in methanol [11] or in SDS micelles [12].

As for many molecules containing hydrocarbon chains, the WAXS signal of lipid bilayers exhibits a distinctive peak with position q 0 ∼ 14 nm -1 , indicative of the packing of these chains in the core of the membrane. Although a full description of the scattered intensity would require an involved model based on liquid state theory [13], the width of the peak provides a quantitative measurement for the positional order of the lipid chains: the longer the range of order, the narrower the peak

The eect of peptide inclusions on the chain peak has been studied for decades [14]. Systematic investigations have shown that some AMPs (e. g. magainin) have a very strong disrupting eect on the local order of the chains: the chain signal disappears almost completely for a modest concentration of inclusions [15,16,17]. With other peptides, the changes in peak position and width are more subtle [18] and can even lead to a sharper chain peak (as for the SARS coronavirus E protein [19]).

To our knowledge, however, no WAXS studies of the eect of GramA on the chain signal have been published. NMR can probe global and local order parameters in various lipid phases and along the lipid chain. Deuterium ( 2 H) NMR has been the method of choice since the 1970s and has proven very successful until today [20,21,22,23]. The eect of GramA on the order parameter of the lipid (or surfactant) chains has already been studied by deuterium ( 2 H) NMR in membranes composed of DMPC [21,24,25,26], DLPC [26] and DDAO [6], but not necessarily at the same temperature, concentration or lipid position as studied here.

Here, we use a novel application of solid-state NMR under magic-angle spinning (MAS) and dipolar recoupling, called the Dipolar Recoupling On-Axis with Scaling and Shape Preservation (DROSS) [27]. It provides similar information as 2 H NMR, by recording simultaneously the isotropic 13 C chemical shifts (at natural abundance) and the 13 C- 1 H dipolar couplings at each carbon position along the lipid or surfactant chain and head group regions. The (absolute value of the) 13

C-1 H orientation order parameter S CH = 3 cos 2 θ -1 /2, with θ the angle between the internuclear vector and the motional axis, is extracted from those dipolar couplings, and the variation of order proles with temperature or cholesterol content has already been probed, with lipids that were dicult to deuterate [28,29]. Using the same approach, we monitor the lipid or surfactant order prole when membranes are doped with dierent concentrations of gramicidin.

The main advantages of 13 C over 2 H are: the possibility to study natural lipids, with no isotopic labeling, and the high spectral resolution provided by 13 C-NMR, allowing the observation of all carbons along the lipid in a single 2D experiment. Segmental order parameters are deduced, via a simple equation, from the doublet splittings in the second dimension of the 2D spectra. The data treatment is simple for nonspecialists and the sample preparation is very easy since there is no need for isotopic enrichment. All these facts make this technique ideal to probe and study new molecules and to be able to compare the results with the ones obtained with other similar particles.

The downsides are the reduced precision in the measurement and the impossibility to extract data from lipids in the gel phase. In particular, carbons at the interfacial region of the lipids (at the glycerol backbone and at the top of the acyl chains) are less sensitive to changes in membrane rigidity, and while subtle changes can be detected with 2 H-NMR, they are dicult to interpret with 13 C-NMR at these positions. Furthermore, the ineciency of the DROSS method in the gel phase would theoretically allow measuring the lipid order in uid phases coexisting with gel phases and quantifying the amount of lipids in each phase. In our measurements, lipids in the gel phase were not abundant enough to be detected.

2 Materials and methods

Sample preparation

The samples were prepared from stock solutions of lipid or surfactant and respectively Gram A in isopropanol. We mix the two solutions at the desired concentration and briey stir the vials using a tabletop vortexer. The resulting solutions are then left to dry under vacuum at room temperature until all the solvent evaporates, as veried by repeated weighing. The absence of residual isopropanol was cheked by

1 H NMR.
We then add the desired amount of water and mix the sample thoroughly using the vortexer and then by centrifuging the vials back and forth. For WAXS, we used a microspatula to deposit small amounts of sample in the opening of a glass X-ray capillary (WJM-Glas Müller GmbH, Berlin), 1.5 or 2 mm in diameter and we centrifuged the capillary until the sample moved to the bottom. We repeated the process until reaching a sample height of about 1.5 cm. The capillary was then either ame-sealed or closed using a glue gun. For NMR, approximately 100 mg of GramA/lipid or GramA/surfactant dispersion in deuterated water were introduced in a 4 mm-diameter rotor for solid-state NMR.

NMR

NMR experiments with DMPC, DLPC and C 12 EO 4 were performed with a Bruker AVANCE 400-WB NMR spectrometer (

1 H resonance at 400 MHz, 13 C resonance at 100 MHz) using a Bruker 4-mm MAS probe. NMR experiments with DDAO were performed with a Bruker AVANCE 300-WB NMR spectrometer ( 1 H resonance at 300 MHz, 13 C resonance at 75 MHz) using a Bruker 4-mm MAS probe. All experiments were performed at 30 • C.

The DROSS pulse sequence [27] with a scaling factor χ = 0.393 was used with carefully set pulse lengths and refocused insensitive nuclei enhanced by polarization transfer (RINEPT) with delays set to 1/8 J and 1/4 J and a J value of 125 Hz. The spinning rate was set at 5 kHz, typical pulse lengths were 13 C (90

• ) = 3 µs, 1 H (90 • ) = 2.
5 µs and 1 H two-pulse phase-modulation (TPPM) decoupling was performed at 50 kHz with a phase modulation angle of 15 • .

1D spectra were acquired using the simple 13 C-RINEPT sequence with the same parameters. For the 2D spectra, 64 free induction decays were acquired, with 64 to 512 scans summed, a recycle delay of 3 s, a spectral width of 32 kHz

and 8000 complex points. The total acquisition time was between 2 and 14 h.

The data were treated using the Bruker TopSpin 3.2 software.

Resonance assignments followed that of previously published data [24,27,22,30,31], using the C ω-n convention, where n is the total number of segments, decreasing from the terminal methyl segment, C ω , to the upper carbonyl seg- ment C 1 . This representation permits a segment-by-segment comparison of the chain regions. Backbone regions are assigned according to the stereospecic nomenclature (sn) convention for the glycerol moiety. Phosphocholine head group carbons are given greek (α, β, γ) letter designations. The internal reference was chosen to be the acyl chain terminal 13 CH 3 resonance assigned to 14 ppm for all lipids and surfactants studied here.

Order parameters were extracted from the 2D DROSS spectra by measuring the dipolar splittings of the Pake doublet at each carbon site. This splitting was converted into a dipolar coupling by taking the scaling factor χ into account. The absolute value of the segmental order parameter is an additional scaling factor χ of the static dipolar coupling into the measured dipolar coupling. Since the static dipolar coupling, on the order of 20 kHz, is not known with high precision for each carbon, we have adjusted it empirically in the case of DMPC, by comparing it to previously determined values [27,22,30].

WAXS

We recorded the scattered intensity I as a function of the scattering vector q = 4π λ sin(θ), where λ is the X-ray wavelength and 2θ is the angle between the incident and the scattered beams.

Lipids X-ray scattering measurements on the GramA/DLPC and GramA/DMPC systems were performed at the ID02 beamline (ESRF, Grenoble), in a SAXS+WAXS conguration, at an X-ray energy of 12.4 keV (λ = 1Å). The WAXS range was from 5 to 53 nm -1 . We recorded the integrated intensity I(q) and subtracted the scattering signal of an empty capillary, as well as that of a water sample (weighted by the water volume fraction in the lipid samples). We used nine peptide-to-lipid molar ratios P/L ranging from 0 to 1/5 and three temperature points: 20, 30 and 40 • C.

The chain peak was tted with a Lorentzian function:

I(q) = I 0 q-q0 γ 2 + 1
.

We are mainly interested in the parameter γ, the half-width at half maximum (HWHM) of the peak.

Surfactants The GramA/DDAO and GramA/C 12 EO 4 systems were studied using an in-house setup using as source a molybdenum rotating anode [32].

The X-ray energy is 17.4 keV (λ = 0.71Å) and the sample-to-detector distance is 75 cm, yielding an accessible q-range of 0.3 to 30 nm -1 . We used ve peptideto-surfactant molar ratios (also denoted by P/L) ranging from 0 to 1/5.5 and eight temperature points, from 0 to 60 • C. The best t for the peak was obtained using a Gaussian function:

I(q) = I 0 exp - (q -q 0 ) 2 2σ 2
. For coherence with the measurements on lipid systems, we present the results in terms of the HWHM γ = √ 2 ln 2 σ.

We emphasize that the dierence in peak shape (Lorentzian vs. Gaussian) is intrinsic to the systems (double-chain lipids vs. single-chain surfactants) and not due to the resolution of the experimental setups, which is much better than the typical HWHM values measured.

3 Results and discussion

NMR

We acquired twelve 2D spectra for various surfactants and GramA concentration. Figure 1 shows the 2D DROSS NMR spectrum of C 12 EO 4 with a molar GramA concentration P/L = 0.118. For each 2D spectrum, slices were extracted at each carbon position and order parameters were deduced. Figure 2 shows a set of such representative slices (at the position C ω-2 ). As already explained, carbons at the glycerol backbone and at the rst two positions along the acyl chains were discarded. Figure 3 shows the order proles determined for each lipid and surfactant, with variable amounts of GramA.

As shown in Figure 3, there is hardly any change for the headgroup region (C α , C β and C γ ), which is expected, considering the high mobility of this region, except in DDAO (CH 3 , C 2 and C 3 ). In the aliphatic region, in DMPC (Figure 3(a)), the order parameter increases for a ratio of P/L = 0.06 and then decreases for the P/L = 0.115. In DLPC and C 12 EO 4 mixtures (Figure 3

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<<=> 89: ' ' ! $" ' !%%# !" !#$ !# !%$ !% ! $ ! & ' ( ) * ' & ' + , - . , ' / 0 , 0 1 . 2 . , ) ) ) ) ) )
)0,3+4'503.5 <?8) 89: ' ' ! @ ' !%%$ A'B'#'''''A'B'%''''A A'B'#'''''A'B'%''''A A'B'#'''''A'B'%''''A C'''''''''''D''''''''''E C'''''''''''D''''''''''E A'B'#'''''A'B'%''''A C'''''''''''D'''''''''''''''''''''''''''''''''''% "'''''''''#''''''''''" Overall, we conclude that the order proles signicantly increase along the acyl chains with the concentration of gramicidin, except in the case of DMPC where the order prole globally increases with the addition of P/L = 0.05 of gramicidin and then decreases at P/L = 0.11. for DDAO where we show that gramicidin has the same eect as on the acyl chains.

Consequently, we show that gramicidin generally rigidies the acyl chains of DLPC, C 12 EO 4 and DDAO, as well as the head group region of DDAO. In the case of DMPC, gramicidin rst rigidies the acyl chains, but more peptide tend to return the membrane to its original uidity.

WAXS

The chain peak has long been used as a marker for the ordered or disordered state of the hydrocarbon chains within the bilayer [34]. For lipids, an important parameter is the main transition (or chain melting) temperature, at which the chains go from a gel to a liquid crystalline (in short, liquid) phase [35].

The main transition temperature of pure DLPC is at about -1 For the lipids, in the liquid phase the peak width increases slightly with P/L for all temperatures (Figure 5). In the gel phase of DMPC at 20 • C (Figure 5 right and Figure 4) this disordering eect is very pronounced, in agreement with the results of several dierent techniques, reviewed in Ref [41] ( V-A). The linear increase in HWHM with P/L can be interpreted as a broadening (rather than a shift) of the transition. The liquid crystalline phase value of the HWHM is reached only at the highest investigated P/L, amounting to one GramA molecule per 5 or six lipids.

For surfactants, which we only studied in the liquid crystalline phase, changes to the chain peak are slight. In C 12 EO 4 membranes, the peak po- sition q 0 decreases very slightly with temperature (Figure 7), while the peak width is almost unchanged by temperature or gramicidin content (Figure 9 right). As an example, we observe a small decrease of q 0 with the temperature at P/L = 0.073 (Figure 7 This conclusion is conrmed by the very modest change in the HWHM values presented in Figure 9 on the right. At P/L = 0, the HWHM is very close to 2.6 nm -1 for all temperatures. As the gramicidin content increases, we observe a small gap between the dierent temperatures: the width stays constant or increases for the lower temperatures (up to about 40 • C) and decreases for the higher ones. This gap widens at high gramicidin content (P /L > 0.07). Fig. 9 HWHM as a function of the concentration P/L, for all measured temperatures. DDAO bilayers (left) and C 12 EO 4 bilayers (right).

In the case of DDAO, the inuence of gramicidin content is more notable than for C 12 EO 4 and the behavior is richer, especially in the presence of choles- terol.

Without cholesterol, the DDAO WAXS peaks coincide for the dierent temperatures at a given inclusion concentration (e.g. in Figure 6 left at P/L = 0.178) whereas the proles dier according to the gramicidin concentration for a given temperature (see Figure 6 right).

These observations dier in presence of cholesterol where for one concentration of gramicidin inclusions (e.g: case of P/L = 0.082 in Figure 8 For the DDAO system, the peak occurs at much lower q 0 with cholesterol than without: q 0 = 12.77 nm -1 at 20 • C, 12.62 nm -1 at 30 • C and 12.28 nm -1 at 50 • C. Thus, the cholesterol expands DDAO bilayers, in contrast with the condensing eect observed in lipid membranes [42,43]. More detailed molecularscale studies would be needed to understand this phenomenon.

Without cholesterol, the width of the main peak in DDAO membranes is little aected by a temperature change, at least between 0 and 60 • C. Without gramicidin, we observe two distinct HWHM values: ∼ 2.38 nm -1 at the lower temperatures (between 0 • C and room temperature) and ∼ 2.5 nm -1 for higher temperatures (between 30 and 60 • C), but this gap closes with the addition of gramicidin, and at high P/L only an insignicant dierence of 0.05 nm -1 persists (Figure 9 left).

On the other hand, at a given temperature the HWHM does vary as a function of P/L. This change is sigmoidal, with an average HWHM of ∼ 2.4 nm -1 for P/L < 0.05 and ∼ 2.7 nm -1 for P/L > 0.11. Thus, above this concentration, the gramicidin decreases slightly the positional order of the chains.

An opposite eect is observed in the presence of cholesterol (Figure 10), where at high temperature (40 -60 • C) the HWHM drops with the P/L: for instance, from 2.37 nm -1 to 2.08 nm -1 at 60 • C. At low temperature (0-30 • C) there is no systematic dependence on P/L.

Overall we can conclude that gramicidin addition has an eect that diers according to the membrane composition. The temperature has a signicant inuence only in the presence of cholesterol. In all surfactant systems and over the temperature range from 0 to 60 • C, the peak is broad, indicating that the alkyl chains are in the liquid crystalline state. There are however subtle dierences between the dierent compositions, as detailed below.

In C 12 EO 4 membranes, the peak position q 0 decreases very slightly with temperature, while the HWHM is almost unchanged by temperature or gramicidin content.

For DDAO (without cholesterol), q 0 also decreases with temperature at a given P/L, but increases with P/L at xed temperature. On adding gram- icidin, the HWHM increases slightly with a sigmoidal dependence on P/L. Thus, a high gramicidin concentration P/L ≥ 0.1 reduces the positional order of the chains in DDAO bilayers.

The opposite behavior is measured in DDAO membranes with cholesterol. Adding gramicidin inclusions have two distinct behaviors depending on the temperature. For low temperatures (between 0 • C and 30 • C) we have a small peptide concentration dependence and a clear temperature correlation, whereas at high temperatures (between 40 • C and 60 • C) we have a strong decrease in the HWHM in presence of inclusions depending only with the P/L content without any variation with the temperature rise. Since at P/L = 0 the HWHM value is very close for the dierent temperatures then we can conclude that adding gramicidin to a membrane containing cholesterol helps rigidify it.

Comparing the NMR and WAXS results

Although the orientational and positional order parameters are distinct physical parameters, one would expect them to be correlated (e.g. straighter molecules can be more tightly packed, as in the gel phase with respect to the uid phase.) This tendency is indeed observed in our measurements, with the exception of DDAO.

We measured by NMR that the orientational order parameter for DMPC increases when adding P/L = 0.05 and slightly decreases at P/L = 0.1 (Figure 3(a)). This behavior was also measured by WAXS for the positional order parameter at both P/L values (Figure 5 right). Similarly, we measured for DLPC acyl chains the same orientational and positional order proles where the order increases for P/L = 0.05 and remains the same when adding P/L = 0.1 gramicidin (Figure 3(b) and 5 left).

As for the C 12 EO 4 surfactant acyl chains, we found a modest raise in both the orientational and the positional order parameters when adding the gramicidin peptide with no dependence on the P/L molar ratio (Figure 3(c) and 9 right).

In the case of DDAO we found that adding gramicidin signicantly increases the orientational order (Figure 3(d)) and decreases the positional order (Figure 9 left). Solid-state NMR also shows an abrupt change in the headgroup region when little GramA is added, followed by a more gradual ordering of the acyl chain when more GramA is added. This may imply a particular geometrical reorganisation of DDAO around the GramA inclusion that could be tested with molecular models.

Conclusions

Using solid-state NMR and wide angle X-ray scattering, we showed that inserting Gramicidin A in lipid and surfactant bilayers modies the local order of the constituent acyl chains depending on multiple factors. In particular, we studied the inuence of membrane composition and temperature on the local order.

The behavior of this local order is quite rich, with signicant dierences between lipids, on the one hand, and single-tail surfactants, on the other, but also between DDAO and all the other systems.

We showed that adding gramicidin inuences the orientational order of the acyl chains and we nd a similar behavior for the orientational order and the positional order, except in the particular case of DDAO.

In this system, GramA content seems to notably inuence the DDAO acyl chains by decreasing their positional order and increasing their orientational order. GramA also inuences the orientational order of the head groups. Also in DDAO, we showed by WAXS that the temperature has a signicant inuence on the positional order only in the presence of cholesterol.

In the gel phase of DMPC, GramA addition leads to a linear decrease in positional order, saturating at the liquid phase value for a molar ratio P/L between 1/6 and 1/5. In the liquid phase, we measure relatively small modications in the local order in terms of position and orientation when adding Gramicidin A, especially in the case of DMPC, DLPC and C 12 EO 4 . This is a very signicant result, which allows further elaboration of elastic models in the presence of inclusions by using the same elastic constants obtained for bare membranes.

As seen above for DDAO, in some membranes the presence of inclusions inuences dierently the positional and orientational order of the acyl chains.

Consequently, combining both techniques (NMR and WAXS) on the same system is very useful in obtaining a full image of the local order. A more detailed analysis could be performed by comparing our results with molecular dynamics simulations. The correlation between changes in the chain order and larger-scale parameters of the bilayer (e.g. the elastic properties) could be established by using dynamic techniques, such as Neutron spin echo.

Fig. 1

 1 Fig. 1 Example of a 2D 1 H -13 C DROSS spectrum for GramA/C 12 EO 4 with P/L = 0.118.

Fig. 2

 2 Fig. 2 Dipolar coupling slices of the C ω- at 30 • C.

FFig. 3

 3 Fig. 3 Orientational order parameter |S CH | for DMPC (a), DLPC (b), C 12 EO 4 (c) and DDAO (d) bilayers embedded with GramA pores for dierent P/L at 30 • C. Error bars are smaller than symbol size.

Fig. 4

 4 Fig. 4 Chain peak for DMPC in bilayers doped with varying amounts of GramA at 20 • C.

Fig. 5

 5 Fig.5 Width of the chain peak for DLPC (left) and DMPC (right) bilayers as a function of the GramA doping at three temperatures.

  left), as well as a very slight increase with P/L at 20 • C, as seen in Figure7right. If we take the overall WAXS peak position shift as a function of temperature and for all inclusions concentration (data not shown) we have a small temperature dependence for each P/L. Comparing the value in absence of inclusion, the peak position slightly shifts after adding gramicidin at a P/L = 0.015 but remains almost the same for the dierent gramicidin content, showing no signicant inuence of the inclusions on the C 12 EO 4 membranes.

Fig. 6

 6 Fig. 6 Scattered signal I(q) for DDAO bilayers, as a function of temperature for the most concentrated sample, with P/L = 0.178 (left) and for all concentrations at T = 40 • C (right).

Fig. 7

 7 Fig. 7 Scattered signal I(q) for C 12 EO 4 bilayers, as a function of temperature for a sample with P/L = 0.073 (left) and as a function of concentration at room temperature: T = 20 • C (right).

8

  Scattered signal I(q) for DDAO/Cholesterol bilayers, as a function of temperature for a sample with P/L = 0.082 (left) and as a function of concentration at T = 50 • C (right).

  left) at dierent temperatures, we observe two families in which the spectra are quasi identical: one group at low temperatures (0-30 • C) and another distinct group at higher temperatures (40 -60 • C). At 20 • C, the peaks for DDAO Cholesterol tend to superpose for P/L > 0.028 (data not shown), whereas at 50 • C (Figure 8 right) the peak proles dier and vary with P/L.

Fig. 10

 10 Fig. 10 HWHM as a function of the concentration P/L, for all measured temperatures in the GramA/DDAO+Cholesterol/ H 2 O system.

  • C[36,37,[START_REF] Cevc | Phospholipid bilayers: physical principles and models[END_REF][START_REF] Ku£erka | [END_REF] and that of pure DMPC is between 23 • C and 24 • C[40,36,37,[START_REF] Cevc | Phospholipid bilayers: physical principles and models[END_REF][START_REF] Ku£erka | [END_REF].

		65				Gram/DMPC 20°C
		60				P/L
		55 50				1/5.21 1/6.26 1/7.82
		45				1/10.43
	I [10 ] -1 mm -3	40 35 30 25				1/14.6 1/21 1/31 1/52 0 (pure DMPC)
		20
		15
		10
		5
		0
		5	10	15	20 q [nm -1 ]	25	30	35

Acknowledgements We thank the CMCP (UPMC, CNRS, Collège de France) for the use of their Bruker AVANCE 300-WB NMR spectrometer. The ESRF is acknowledged for the provision of beamtime (experiment SC-2876) and Jérémie Gummel for his support. This work was supported by the ANR under contract MEMINT (2012-BS04-0023). We also acknowledge B. Abécassis and O. Taché for their support with the WAXS experiment on the MOMAC setup at the LPS.


Bastien Confais

Adrien Lebre

Benoît Parrein

Improving locality of an object store working in a Fog environment

Introduction

The Cloud Computing model relying on few datacenters located far from the users cannot satisfy the new constraints of the Internet of Things (IoT), especially in terms of latency and reactivity. Fog and Edge computing infrastructures have been proposed as an alternative [START_REF] Bonomi | Fog Computing and Its Role in the Internet of Things[END_REF]. This new paradigm consists of deploying small data-centers close to the users located at the edge to provide them low-latency computing. Figure 1 shows that the Fog platform is composed of a significant number of that can be geographically spread over a large area. IoT devices and users are mobile and are connected to the closest site of Fog Computing. We consider the network latency between the client and the closest Fog site to be lower than the network latency between the Fog sites.

We work on a storage solution for this infrastructure. Our goal is to create a seamless storage experience towards different sites, capable of working in a disconnected mode by containing the network traffic to the sites solliciting the data. We previously proposed to use the object store Interplanetary FileSystem [START_REF] Benet | IPFS -Content Addressed, Versioned, P2P File System[END_REF][START_REF] Confais | Performance Analysis of Object Store Systems in a Fog and Edge Computing Infrastructure[END_REF] as a Fog storage system because it enables users and IoT devices to write locally on their site but also to automatically relocate the accessed objects on the site they are requested. After improving the locality of IPFS by adding a Scale-Out NAS [START_REF] Confais | An object store service for a Fog/Edge Computing infrastructure based on ipfs and a scale-out NAS[END_REF] on each site and proposing to manage location of object in a tree, we evaluated our proposals on the Grid'5000 system by adding virtual network latencies between Fog nodes and clients. We now want to evaluate it in a more realistic environment.

We propose to evaluate the performance of the storage system using the couple G5K/FIT. The G5K platform emulates the Fog layer whereas the Edge layer is emulated on the FIT one. Locations like Grenoble, Lyon or Lille are appropriate for our experiment because they host both a G5K and a FIT site, so that we can expect the network latency between IoT nodes and Fog nodes to be low. Grid'5000 and the FIT platforms. Figure 2 shows the general topology used for the experiment. We developed a RIOT application to enable IoT nodes to access an IPFS server. The scenario consists in reading a value from a sensor, and storing it in an object of the IPFS server located on the Fog. The challenge for a such scenario is to connect the two platforms. Reaching the G5K platform from the IoT node is not easy, because G5K nodes and FIT nodes are connected to the Internet through IPv4-NAT and the G5K platform does not provide any IPv6 connectivity. Because of the lack of end-to-end IP connectivity between the two platforms, we encapsulate messages in SSH tunnels between a A8-node on the FIT platform and the IPFS node. Instead of accessing directly the IPFS node (plain arrow), the client accesses it from the introduced A8-node that acts like a proxy (dashed arrow). This solution works but is not ideal not only because the tunnel degrades the performance but also because the IP routing is not direct and the path from the two platforms in Grenoble goes through Sophia, increasing the network latency.

Conclusion

This experiment was one of the first involving simultaneously the two platforms and we pointed out the difficulties to interconnect them. Once these problems will be solved, we plan to perform a more advanced scenario involving node mobility or the possibility for a node to choose the Fog site to use.

Fig. 1 :

 1 Fig. 1: Overview of a Cloud, Fog and Edge infrastructure.

Fig. 2 :

 2 Fig.2: General architecture of our experiment and interconnection between the Grid'5000 and the FIT platforms. Figure2shows the general topology used for the experiment. We developed a RIOT application to enable IoT nodes to access an IPFS server. The scenario consists in reading a value from a sensor, and storing it in an object of the IPFS server located on the Fog. The challenge for a such scenario is to connect the two platforms. Reaching the G5K platform from the IoT node is not easy, because G5K nodes and FIT nodes are connected to the Internet through IPv4-NAT and the G5K platform does not provide any IPv6 connectivity. Because of the lack of end-to-end IP connectivity between the two platforms, we encapsulate messages in SSH tunnels between a A8-node on the FIT platform and the IPFS node. Instead of accessing directly the IPFS node (plain arrow), the client accesses it from the introduced A8-node that acts like a proxy (dashed arrow). This solution works but is not ideal not only because the tunnel degrades the performance but also because the IP routing is not direct and the path from the two platforms in Grenoble goes through Sophia, increasing the network latency.


Guilherme Machado
email: guilherme.machado@imag.fr

Arthur Stricher
email: arthur.stricher@insa-lyon.fr

Grégory Chagnon
email: gregory.chagnon@imag.fr

Denis Favier
email: denis.favier@imag.fr

Mechanical behavior of architectured photosensitive silicone membranes: experimental data and numerical analysis

Keywords: architectured membranes, photosensitive silicone, biocompatible silicone, bulge test, bi-material hyperelastic solid

Introduction

Facing increasing demands for multifunctional solutions, architectured materials take an increasingly important place in many applications in order to design specific mechanical properties for a given purpose. Very often the function is not provided by the local property only (grain size, precipitation, polymer chain design and interchain bonding, state of crystallization), but by the interplay between the shape, the properties, and possible association of materials [START_REF] Brechet | Embury Architectured materials: Expanding materials space[END_REF].

For instance, some polymers are architectured due to their chain design [START_REF] Zuo | Effects of block architecture on structure and mechanical properties of olefin block copolymers under uniaxial deformation[END_REF]. Usually, this strategy operates at scales between 1 nm and 10 µm. Architectured silicone materials can also be composed with an association of materials (also called hybrid material) for example fiberreinforced [START_REF] Bailly | In-plane mechanics of soft architectured fibre-reinforced silicone rubber membranes[END_REF] or NiTi-reinforced membranes [START_REF] Rey | An original architectured NiTi silicone rubber structure for biomedical applications[END_REF]. Even if such composites are very good candidates for biomimetic membranes, this solution involves the integration of different synthetic materials often associated with local mismatches of mechanical properties and adhesion difficulties. Local mismatches may cause excessive stress concentrations within the structure and thus premature failure of the composite upon stretching. Thus, one of the challenges of the architectured material is in ensuring efficient stress transfer and in avoiding local failure between regions of different mechanical properties.

Other materials can be considered as architectured because of their geometry. [START_REF] Meunier | Anisotropic large deformation of geometrically architectured unfilled silicone membranes[END_REF] and [START_REF] Rebouah | Development and modeling of filled silicone architectured membranes[END_REF] developed crenelated membranes with an unfilled and filled silicone rubber. The main advantage of these membranes is that they present an anisotropic behavior without any interface in the material. Indeed, the crenels and their orientations allow to induce and control the anisotropy, but this fact is limited by the out-plane geometry and the process to obtain the reinforced membrane.

This paper focuses on the mechanical behavior of architectured silicone membranes where the membrane architecture is controlled by the in-plane intrinsic properties but also by a desired topology at scale between the microstructure and the application. The concept is to create a heterogeneous material with locally tuned mechanical properties by changing the local crosslink density. The approach can be exploited, for example, to create bioinspired membranes that mimic anisotropic structural properties of soft tissues. In this context, Section 2 presents all precautions concerning the experimental mechanical testing procedures and strain field measurements techniques. Experimental data and analyzes are presented into two parts. First, in Section 3, the three deformations modes (uniaxial, planar and equibiaxial tensile tests) for each phase are independently tested. Second, in Section 4, the bulge test of graded membrane containing two phases. In Section 5, a finite element analysis (FEA) is carried out using a hyperelastic model fitted simultaneously on the three previous tensile tests. Then, the numerical model was used to try to predict the bimaterial and results are discussed. Finally, Section 6 contains some concluding remarks and outlines some future perspectives.

Testing procedures background and strain field measurements techniques

A series of mechanical tests were carried out to characterize the silicone mechanical behavior in its soft and hard phases. First, for the three deformations modes: uniaxial, planar (pure shear) and equibiaxial tensile tests; second, for the bulge test of graded membrane containing two phases.

Preparation of the silicone specimens

Samples were prepared in the IMP laboratory (Ingénierie des Matériaux Polymères -Villeurbanne, France), using the polydimethylsiloxane (PDMS) elastomer in addition of an UV-sensitive photoinhibitor. The membrane was selectively exposed to UV radiation then the cross-linking of the UV exposed elastomer is inhibited, leading to a softer material than the unexposed zone.

From this point forward, the soft phase denotes the UV exposed material and the hard phase the unexposed one.

In-plane tests

In-plane quasi-static experiments were conducted on a Gabo Explorer testing machine with ±25 N and ±500 N load cells for uniaxial tension and planar tension respectively. A 2D digital image correlation system (DIC) was used during the test to obtain 2D fields at the surface of plane specimens. The commercial VIC-2D 2009 software package from Correlated solutions was used to acquire images. The images were recorded at 1 Hz with a Pike F-421B/C CCD camera with a sensor resolution of C r = 7.4 µm/pixel. The reason for this large sensor format is the goal to achieve high resolution images with low noise. The 50 mm camera lens was set to f /22 using a 50 pixels extension ring. Grayscale 8 bit images were captured using a full scan of 2048 pixels × 2048 pixels. After all, a cross-correlation function was used and displacement vectors were calculated by correlating square facets (or subsets) of f size = 21 pixels and grid spacing G s = 10 pixels to carry out the correlation process for the undeformed and deformed images. To achieve a sub-pixel accuracy, optimized 8-tap splines were used for the gray value interpolation. As the optimization criteria for the subset matching, a zero-normalized squared difference was adopted, which is insensitive to offset and scale in lighting. For the weighting function, the Gaussian distribution was selected, as it provides the best compromise between spatial and displacement resolution [START_REF] Sutton | Image correlation for shape, motion and deformation measurements[END_REF]. In uniaxial tensile experiment, the spatial resolution (the physical size per pixel) was S r = 15 µm. Likewise, for the planar tension S r = 7 µm.

Out-plane tests

The bulge test was conducted in order to determinate an equibiaxial state of both phases and also tested the soft-hard bimaterial. A syringe driver was used and the internal pressure is measured by an AZ-8215 digital manometer. Seeing that the material is partially transparent, milk was used as hydrostatic fluid to increase the gray contrast for DIC and to avoid internal reflections. Inflation was sufficient slow to obtain a quasi-static load. Under the assumption of material isotropy over the circumferential direction, the principal directions of both stretch and stress tensors at each material particle are known ab initio to be the meridional and circumferential directions of the membrane surface. From this point forward, these directions will be denoted by the subscripts m and c respectively.

Assuming quasi-static motion, the equilibrium equations for a thin axisymmetric isotropic membrane, as adopted by Hill [START_REF] Hill | A theory of the plastic bulging of a metal diaphragm by lateral pressure[END_REF], can be expressed as

σ m = p 2h κ c (1)
σ c = p 2h κ c 2 - κ m κ c (2)
where (σ m , σ c ) are the meridional and circumferential stresses and (κ m , κ c ) are the meridional and circumferential curvatures. h is the current thickness and p is the time-dependent normal pressure acting uniformly (dp/dR = 0) over the radius R. As mentioned in [START_REF] Wineman | Large axisymmetric inflation of a nonlinear viscoelastic membrane by lateral pressure[END_REF] and [START_REF] Humphrey | Computer methods in membrane biomechanics[END_REF] a remarkable consequence of membrane theory is that it admits equilibrium solutions without explicitly requiring a constitutive equation, since the equilibrium equations are derived directly by balancing forces of a deformed element shape. As a consequence, they are valid for all classes of in-plane isotropic materials.

Recently, Machado et al. [START_REF] Machado | Membrane curvatures and stress-strain full fields of axisymmetric bulge tests from 3D-DIC measurements. Theory and validation on virtual and experimental results[END_REF] presented a methodology to compute the membrane curvature of the bulge test from 3D-DIC measurements. A very convenient calculation scheme was proposed based on the surface representation in curvilinear coordinates. From that scheme, the circumferential and meridional curvatures, and also the respective stresses, can be computed. In [START_REF] Machado | Membrane curvatures and stress-strain full fields of axisymmetric bulge tests from 3D-DIC measurements. Theory and validation on virtual and experimental results[END_REF] authors presented an evaluation scheme for the bulge test based on the determination of the surface curvature tensor and the membrane stress tensor. With this method, the circumferential as well as the meridional stress can be determined at every stage and position of the specimen.

The commercial VIC-3D 7 software package from Correlated solutions was used to acquire images using two digital Pike cameras described in Section 2.3. Both cameras were set up at D = 150 mm distance and 35 • angle to the specimen using 28 mm focal length lenses opened at f /16. Previous to the test, a good calibration of the 3D-DIC system is required. The following correlation options were chosen: 8-tap splines were used for the gray value interpolation, zeronormalized squared difference for the subset matching and the Gaussian distribution for the weighting function. Square facets of f size = 15 pixels and grid spacing G s = 5 pixels to carry out the correlation process for the undeformed and deformed images. The obtained spatial resolution is S r = 15 µm.

Note that the spatial resolution (S r ) of the discretized surface depends essentially on camera sensor resolution (C r ) and on the choice of the grid spacing (G s ) that defines the distance between the data points on the object. The grid spacing is the distance between the grid points in pixel. Thus, grid spacing limits the spatial resolution, as each grid point represents one single data point of the result. The facet size controls the area of the image that is used to track the displacement between images. The minimal facet size (f size ) is limited by the size and roughness of the stochastic pattern on the object surface. Each facet must be large enough to ensure that there is a sufficiently distinctive pattern, with good contrast features, contained in the area-of-interest used for correlation.

Soft and Hard phases: experimental results and analysis

Uniaxial tension test

Uniaxial tensile tests were performed on small dog-bone shaped specimens. The samples had an initial gage length l 0 = 12 mm, width w 0 = 4 mm and thickness h 0 = 0.8 mm. During the test, using an elongation rate of λ = 3.0 × 10 -2 s -1 , the nominal stress tensor P (First Piola-Kirchhoff stress tensor) is assumed to be homogeneous within the gauge region as well as the deformation gradient tensor F. Since the current thickness is not measured, the material is assumed to be incompressible, i.e., det (F) = 1. A cyclic loading-unloading test was realized for soft and hard phases, the curves are presented in Fig. 1. In the same figure, the first load of both phases are plotted. Different phenomena are highlighted, first a large stress-softening appears by comparing the two first loading at each strain level. A little hysteresis after the first cycle is observed. Moreover, few residual elongation is observed for both phases. for soft and hard phases.

Planar tension test

The pure shear strain state was approached by performing planar tension test. The initial height l 0 , the constant width w 0 and the thickness h 0 of samples were 4.5 mm, 40 mm and 0.8 mm, respectively. The width of the specimen used for planar tension test must be at least ten times greater than its length. These dimensions have as objective to create an experiment where the specimen is constrained in the lateral direction such that all specimen thinning occurs in the thickness direction. A cyclic planar loading test was realized for both phases at λ = 1.0 × 10 -2 s -1 . The results are presented in Fig. 2. Planar tensile response, likewise uniaxial traction, presents the same phenomena. For the soft phase, the maximum principal stretch experienced by the planar specimens are smaller if compared with uniaxial tensile test specimens.

In general, this limitation lies in the fact that the planar tensile specimens must be constrained in the lateral direction without slipping. In this manner, the annoying premature tearing at the grips is observed. This is the major difficulty in planar tensile tests of thin specimens.

Equibiaxial tension using the bulge test

The equibiaxial tension state is approached by the bulge test. Due to the axial-symmetry of the experimental configuration the equibiaxiality of the stress and strain is obtained at the top the inflated sample. The elongation rate was not controlled, but the pressure p is slowly increased. The stress-strain curve for the central area are presented in Fig. 3 for a cyclic loading.

The response are qualitatively similar to uniaxial loading with hysteresis and stress-softening.

Analysis

Table 1 presents a comparative for soft and hard phases for the three different experimental load cases (uniaxial, plane shear and equibiaxial). It is easily determined from classical isotropic elasticity theory that, during quasi-linear stages, the ratio of the stress-strain slopes between uniaxial tension (E) and equibiaxial tension (

E e ) is E e /E = 1/(1 -ν).
The ratio E e /E was determined experimentally to be 2 for both phases. This is compatible with an incompressible material with Poisson's ratio ν = 0.5. The Young's modulus ratio between the two phases (E H/S ) was about 3.5 for uniaxial and equibiaxial states and 2.5 for plane shear. At the beginning of loading, the stress ratio between phases at λ = 1.1 of strain (P 10%

S

) are closer than stress ratio at λ = 2.0. The mean stress ratio P calculates over all load history is kept practically constant for all loading cases. The specimen disk effective dimensions are 18.5 mm of radius and 0.4 mm of thickness. The UV-irradiated zone is concentric circle of 10 mm diameter. Cross-linking of the UV exposed elastomer is inhibited, leading to softer region than the surrounding unexposed part,as illustrated in Fig. 4.

The bulge test was chosen to test the bimaterial for two main reasons: stress-concentrations can be easily access the soft-hard interface is far from boundary conditions; each inflation state involves a heterogeneous stress-strain state which can be determined analytically. Having said that, bulge offers a valuable data for modeling benchmark. Inflations were performed from 1 kPa to a maximum pressure of 25 kPa, therefore, for clarity, three levels were chosen to present the results: 6, 15 and 25 kPa. These inflations yielded principal stretches at the pole of about 1.16, 1.43 and 2.21 respectively. Fig. 6 presents principal stretches values (λ m , λ c ) obtained from 3D-DIC system. Save the pole (R = 0) and the clamped boundary (R = 1), all material points involve a heterogeneous strain state. As expected, the circumferential stretch λ c tends to one, i.e., a pure planar stretching behavior towards the clamped boundary (R → 1). However, most of the hard phase deformation is on the circumferential direction since λ c is less than 1.2 even for the maximal pressure level. Principal curvatures (κ m , κ c ) and principal stresses (σ m , σ c ) were computed as explained in [START_REF] Machado | Membrane curvatures and stress-strain full fields of axisymmetric bulge tests from 3D-DIC measurements. Theory and validation on virtual and experimental results[END_REF]. Fig. 7 shows the experimental curves. With respect to principal curvature distributions, note that equibiaxial membrane deformations near the membrane pole (R = 0) are associated with an approximately spherical geometry, i.e., κ m ≈ κ c . The small difference may be explained by the fact that the umbilical point may not lie exactly on the Z direction axis. Note that for all pressure levels, the meridional curvature κ m presents an inflection point representing changes from convex to concave curvature on the soft-hard interface (R = 0.27).

•• • • •• • • •• • • •• • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • •• • • •• •••• • • • ••••••• • •• • • • (b)
With regard to the stress plots in Figs. 7c and7d, the stress state can be assumed to be equibiaxial at the pole (R = 0). Both stresses, σ m and σ c , experience an increasingly upward turn for R 0.27 when pressure increases. Two inflections points can be observed around R = 0.27 and R = 0.36 in all directions for all pressure loads.

Finite element simulations

In the previous section, bulge test was used to obtain stress and strain fields for a nonhomogeneous material. Stress were calculated without explicitly specifying a constitutive relation for the material. The aim of this section is to compare these results with usual finite element analysis using the classical Mooney-Rivlin hyperelastic constitutive equation. Stress-softening and hysteresis are not regarded, thus only the first loading behavior was investigated.

Hyperelastic fitting using the Mooney-Rivlin hyperelastic model

Assuming an incompressible isotropic hyperelastic material behavior, the two parameters Mooney-Rivlin model is expressed as

W Ī1 , Ī2 = C 10 Ī1 -3 + C 01 Ī2 -3 (3)
where C 10 and C 01 are material parameters that must be identified. Ī1 and Ī2 are the first and second strain invariants of the the isochoric elastic response. Due to its mathematical simplicity as well as its prediction accuracy in the range of moderately large strains, Mooney-Rivlin model has been widely employed in the description of the behavior of rubbery materials. It is known that different deformation modes are required to obtain the parameters that define the stressstrain relationship accurately. Uniaxial, planar and equibiaxial data acquired in Section 3 are simultaneously involved in a least square minimization in order to extract the sets of material parameters for each material phase. Table 2 summarizes the two sets of C 10 and C 01 parameters. Fig. 8 shows the stress-strain curves of the first loading experimental data and the Mooney-Rivlin model fitting for each deformation mode. The adopted fitting procedure allows a material model description that is valid for a general deformation state. As expected, the model shows a good agreement with uniaxial and planar tensile tests data up to 100% of strain, i.e., λ = 2.0.

Moreover, the Mooney-Rivlin model starts to fail to account strains larger than λ = 1.3 for the equibiaxial tensile test, in particular for the hard phase. Nevertheless, as pointed out in [START_REF] Marckmann | Comparison of hyperelastic models for rubberlike materials[END_REF],

there are very few hyperelastic constitutive models able to simultaneously simulate both the multi-dimensional data with an unique set of material parameters.

Experimental and numerical comparison of bulge test results

The non-homogeneous bulge test was simulated by an axisymmetric using Abaqus commercial finite element code. Continuum eight-node biquadratic hybrid fully integrated elements (CAX8H) were used. Based on the result of the mesh sensitivity study, the optimal global element size for membrane mesh was h = 35 µm. At the soft-hard interface, a non-adaptive h-refinement was used to improve mesh quality employing a mesh size gradient used only on the R direction, resulting an element edge size about h/6 over the interface neighborhood. Over the membrane thickness, 30 Gauss integration points were used.

Results of FEA using the Mooney-Rivlin model are superposed with the experimental fields (principal stretches and principal stresses) in Fig. 9. Two pressure levels were chosen to present the results: 6 and 25 kPa. All numerical predictions follow qualitatively the trends of the experimental data. It is possible to observe a discontinuity in the model response at R = 0.27 even if the an h-refinement was used in this zone.

Considering the principal stresses plot in Figs. 9c and9d, numerical simulations do not correspond well with experimental ones in both load cases. This result can be related to the limitations of the Mooney-Rivlin model to fit complex stress-strain states at large strain levels.

Fig. 10a presents the FEA errors (e m , e c ) with respect to principal stresses over the bulge profile in both deformed configurations. Using a confidence interval of 95%, the mean error are êm = 31%, êc = 30% for the lower pressure level; and êm = 35%, êm = 29% for the highest pressure level. Regardless of the soft-hard interface, stress discrepancies are independent of the deformation level. This fact is also observed in Fig. 10b where the deviations of the stress ratio σ m /σ c are very close for both load levels.

Modeling analysis

Presented results show the limitations of the classical finite element method to tackle heterogeneous systems with moderate modulus mismatch across the material interface undergoing large strains, with an incompressible non-linear hyperelastic material behavior. Results reveals that a more sophisticate representation of the soft-hard interface must be taken into account by numerical modeling. For example, [START_REF] Srinivasan | Generalized finite element method for modeling nearly incompressible bimaterial hyperelastic solids[END_REF] proposed an extension of the generalized finite element method to tackle heterogeneous systems with non-linear hyperelastic materials. However, it must be recognized that outside the context of this study. Independently of the numerical treatment, a more detailed knowledge of the influence the material interface on the macroscopical mechanical behavior is necessary. For this, considering the Mooney-Rivlin model in Eq. 3 in terms of the principal stretches

σ m = 2 C 10 (λ m ) 2 - 1 λ m λ c 2 + C 01 1 λ m λ c 2 -(λ m ) 2 (4)
σ c = 2 C 10 (λ c ) 2 - 1 λ m λ c 2 + C 01 1 λ m λ c 2 -(λ c ) 2 . (5)
Replacing the measured principal stretches in Eqs. 4 and 5 and the previous identified parameters Finally, keeping in mind the experimental spatial resolution of 15 µm, the phase transition is estimated to be about 3.15 mm, i.e., 17% total sample radius. Within the tested loading range, the size of soft-to-hard transition can be assumed independent of the stress-strain level.

F i t o f C 0 1 C 1 0 F i t o f C 1 0 R C 1 0 , C 0

Conclusions

Results show the mechanical behavior of photosensitive silicone membranes with a variable set of mechanical properties within the same material. With a reversible in-plane stretchability up to 200%, the soft-to-hard transition was expressed by a factor 3.57 in the Young's modulus within a single continuous silicone membrane combined with a mean stress factor about 2.5 times.

The approach was tested using the bulge test and the presented results using a bimaterial are distinct from previous investigations of the classic circular homogeneous membrane inflation problem. The mechanical response of the soft-hard interface was observed by inflections on the principal curvatures fields along the meridional and circumferential directions. Analysis of the stress distribution throughout the meridional-section of the membrane revealed low stress peaks at soft-to-hard transition. The results demonstrate that under high strains levels no macroscopic damage was detected. The local cross-linking control eliminates the interfaces between different materials, leading to heterogeneous membrane with efficient stress transfer throughout the structure.

The numerical investigation provided information on the respective contributions of each material phase on its effective behavior under the inflation. Therefore, presented results show the limitations of the classical finite element method to tackle heterogeneous systems with moderate modulus mismatch across the material interface undergoing large strains, with an incompressible non-linear hyperelastic material behavior. Using the experimental local stress-strains values, it was possible to characterize the macroscopic influence of the soft-to-hard interface with spatial resolution of 15 µm. Later, a more sophisticated numerical strategy can be used to describe the soft-hard interface and then the graded membrane global behavior, based on the presented results. Further work to create, test and optimize more complex architectures, is ongoing using the experimental approaches described in the present paper.

Figure 1 :

 1 Figure 1: Nominal stress-strain curves resulting from cyclic loading-unloading tensile test at λ = 3.0 × 10 -2 s -1 for soft and hard phases.

Figure 2 :

 2 Figure 2: Nominal stress-strain curves resulting from cyclic loading-unloading planar tension test at λ = 1.0 × 10 -2 s -1 for soft and hard phases.

Figure 3 :

 3 Figure 3: Nominal stress-strain curves resulting from cyclic loading-unloading equibiaxial test: (a) hard phase; (b) soft phase.

Figure 4 :

 4 Figure 4: The photosensitive material sample and the bulge test configuration. (See online version for color figure.)

Figure 5 :

 5 Figure 5: Bulge test setup using 3D-DIC technique. (a) Experimental image superposed with the Green-Lagrange major principal strain field (meridional strain); (b) Profiles of the inflated membrane composed by soft and hard phases for different pressure loads. (See online version for color figure.)

Figure 6 :

 6 Figure 6: The strain distribution of a deformed foil vs. normalized radius of the circular membrane. (a) meridional direction λm; (b) circumferential direction λc.

Figure 7 :

 7 Figure 7: Distributions of principal direction of experimental fields corresponding to three different inflation states: (a) meridional curvature κm; (b) circumferential curvature κc; (c) meridional Cauchy stress σm; (c) circumferential Cauchy stress σc.

Figure 8 :

 8 Figure 8: Experimental data for soft and hard phases and the hyperelastic fitting using the Mooney-Rivlin (MR) hyperelastic model: (a) Uniaxial (b) Planar and (c) equibiaxial tensile tests.

Figure 9 :

 9 Figure 9: Principal stretches (λm, λc) and Cauchy stress (σm, σc) confronted with the finite element analysis (FEA), corresponding to 6 kPa and 25 kPa inflation states.

Figure 10 :

 10 Figure 10: (a) FEA errors with respect to principal stresses in both deformed configurations; (b) Principal stress ratio (σm/σc) confronted with the finite element results (FEA).

CFigure 11 :

 11 Figs.11a and 11bshows a good agreement between the experimental (Exp) and Mooney-Rivlin (MR) stresses for both inflation states.In other to determinate the macroscopic influence of the soft-to-hard transition the parameters C 10 and C 01 were evaluated using the Eqs. 4 and 5 using the the measured principal stretches and the experimental stresses obtained by Eqs. 1 and 2, using the different directions and different load levels experimental information. The same ratios C 10 /C 01 of soft and hard phases from previous identification (Table2) were kept. Thus, one obtains a description of the spatial distribution of these parameters, as presented in Fig.12a. It is possible to observe that the soft-to-hard transition transition is almost symmetric with respect the position R = 0.27 and the material parameters gradient extends over the R = [0.21, 0.38] interval.The Mooney-Rivlin (MR) stresses in Eqs. 4 and 5 were recalculated, but now using the functions C 10 (R) and C 01 (R), fitted on experimental results using a sigmoid function. Results

1 (Figure 12 :

 112 Figure 12: (a) Mooney-Rivlin parameters gradient obtained using the the experimental local stress-strain states. (b) Errors with respect to principal stresses in both deformed configurations using the Mooney-Rivlin with a material parameters gradient over the R = [0.21, 0.38] interval.

Table 1 :

 1 A comparative for soft and hard phases for the three different experimental loading cases.

	Parameter		Unit Uniaxial Plane shear Equibiaxial
	Hard phase elastic modulus	E H	MPa	2.50	3.50	4.90
	Soft phase elastic modulus	E S	MPa	0.70	1.40	1.40
	Elastic modulus ratio	E H/S	-	3.57	2.50	3.50
	Hard stress at 10% Soft stress at 10%	P 10% H P 10% S	MPa MPa	0.24 0.08	0.31 0.13	0.45 0.13
	Hard stress at 100% Soft stress at 100% Mean stress ratio	P 100% H P 100% S P	MPa MPa -	1.43 0.61 2.44	1.67 0.68 2.44	1.93 0.85 2.90
	4. Bulge test with soft-hard phase sample

Table 2 :

 2 Fitted parameters (in MPa) of the Mooney-Rivlin constitutive equation for soft and hard phases.

	Parameters hard soft
	C 10	0.35 0.18
	C 01	0.10 0.01

Acknowledgment

The authors wish to acknowledge the financial support of the French ANR research program SAMBA: Silicone Architectured Membranes for Biomedical Applications (Blanc SIMI 9 2012).

We thank Laurent Chazeau, Renaud Rinaldi and Franois Ganachaud for fruitful discussions.