Marie-Noëlle Labour

Amélie Banc

Audrey Tourette

Frédérique Cunin

Jean-Michel Verdier

Jean-Marie Devoisselle

Anne Marcilhac

Emmanuel Belamie
email: emmanuel.belamie@enscm.fr

Thick collagen-based matrices including growth factors to induce 3D neural differentiation

Keywords: Neural differentiation, compartmentation, 3D scaffold, growth factor, coupling

Designing synthetic microenvironments for cellular investigations is a very active area of research at the crossroad of cell biology and materials science. The present work describes the design and functionalization of a 3D culture support dedicated to the study of neural cells differentiation. It is based on a dense self-assembled collagen matrix stabilized by 100 nmwide interconnected native fibrils, without chemical cross-linking. The matrices were made suited for cell manipulations and direct observation in confocal microscopy by anchoring them to traditional glass supports, with a calibrated thickness of c.a. 50 µm. The matrix composition can be readily adapted to specific neural cell types, notably by incorporating appropriate neurotrophic growth factors. PC-12 and SH-SY5Y lines both exhibit high differentiation rates (up to 66% and 49% respectively) with the corresponding growth factors (500 ng NGF, 50 ng BDNF) simply impregnated and slowly released from the support. We also show that both growth factors can be chemically conjugated (EDC/NHS) throughout the matrix and yield similar differentiation rates (61% and 52% respectively). Finally, we observed neurite outgrowth over several tens of microns within the 3D matrix, with both diffusing and immobilized growth factors, which raises the possibility of designing simplified brain tissue model materials.

1-Introduction

Cellular microenvironments, including the structure and rigidity of the substrate, topological and chemical cues like growth factor concentration, orient the cell faith and behavior in physiological conditions. Smart devices for in vitro investigations of cellular events involved in their functions and in pathological situations must include a set of these stimuli associated to a spatio-temporal control of their application to the cells [START_REF] Lutolf | Designing materials to direct stem-cell fate[END_REF][START_REF] Perez-Castillejos | Replication of the 3D architecture of tissues[END_REF]. The response to such cues gives an opportunity to orient the behavior of cells, favor a particular phenotypic engagement, and study specific aspects of their functions and dysfunctions. More specifically, a very active area of research is dedicated to the study and control of neural cell differentiation, and in particular the outgrowth and guidance of neurite, and the effects of distal events on the whole cell faith. Typical examples are the retrograde signaling triggered by ligand-receptor interactions on distant neurites [START_REF] Macinnis | Retrograde support of neuronal survival without retrograde transport of nerve growth factor[END_REF][START_REF] Campenot | Retrograde transport of neurotrophins: fact and function[END_REF], or Aβ-induced neurite degeneration followed by cell death [START_REF] Song | Apoptosis is secondary to non-apoptotic axonal degeneration in neurons exposed to Aβ in distal axons[END_REF]. Systems developed to this purpose most often rely on microfabrication techniques to design patterns with adhesion molecules or growth factor gradients [START_REF] Kim | Biomimetic nanopatterns as enabling tools for analysis and control of live cells[END_REF][START_REF] Rosoff | Generating controlled molecular gradients in 3D gels[END_REF][START_REF] Cao | Defining the concentration gradient of nerve growth factor for guided neurite outgrowth[END_REF][START_REF] Gomez | Immobilized nerve growth factor and microtopography have distinct effects on polarization versus axon elongation in hippocampal cells in culture[END_REF][START_REF] Yeung | Modulation of the growth and guidance of rat brain stem neurons using patterned extracellular matrix proteins[END_REF][START_REF] Liu | A Microfluidic Chamber for Analysis of Neuron-to-Cell Spread and Axonal Transport of an Alpha-Herpesvirus[END_REF][START_REF] Bechara | Template synthesized poly(epsilon-caprolactone) nanowire surfaces for neural tissue engineering[END_REF]. Such patterned substrates are useful to study the formation and branching of neurites, but also axonal dystrophies notably involved in neurodegenerative pathologies like Alzheimer disease (AD).

Several compartmented systems have also been developed with the aim of studying the response of neurites separately from that of the rest of the cell [START_REF] Campenot | Local control of neurite development by nerve growth factor[END_REF][START_REF] Hayman | Enhanced neurite outgrowth by human neurons grown on solid three-dimensional scaffolds[END_REF][START_REF] Brayfield | Excimer laser channel creation in polyethersulfone hollow fibers for compartmentalized in vitro neuronal cell culture scaffolds[END_REF][START_REF] Taylor | Microfluidic Multicompartment Device for Neuroscience Research †[END_REF][START_REF] Taylor | A microfluidic culture platform for CNS axonal injury, regeneration and transport[END_REF][START_REF] Taylor | Microfluidic local perfusion chambers for the visualization and manipulation of synapses[END_REF][START_REF] Cohen | Neurotrophin-mediated dendrite-tonucleus signaling revealed by microfluidic compartmentalization of dendrites[END_REF][START_REF] Steward | Demonstration of local protein synthesis within dendrites using a new cell culture system that permits the isolation of living axons and dendrites from their cell bodies[END_REF]. However, in most cases, the cells mostly evolve in a 2D environment and/or the set-up does not permit the vizualization in 3D of cell protrusions like neuritic extensions.

Here, we have developed a 3D biomimetic compartmented system suited for the study of cellular events occurring within the matrix, near the tip of neurites. Macromolecules of the extracellular matrix (ECM) like collagen, fibronectin or laminin are choice substrates to enhance cell adhesion, and in some cases are required for the proper differentiation of specific cell types [START_REF] Nakajima | Combinatorial protein display for the cell-based screening of biomaterials that direct neural stem cell differentiation[END_REF][START_REF] Singh | Biomaterials for stem cell differentiation[END_REF]. Most often these adhesion molecules are deposited as a very thin layer onto the surface of glass or plastic dishes. The main drawbacks of such very thin coatings are that i) the mechanical properties sensed by the cells are those of the underlying material (glass, plastic), which can largely modify the cells behavior [START_REF] Engler | Substrate compliance versus ligand density in cell on gel responses[END_REF] and ii) the study of cellular events is mostly limited to two dimensions. Conversely, cells seeded on top of thick hydrogels comprised of either polysaccharides or proteins like collagen [START_REF] Besseau | Production of ordered collagen matrices for three-dimensional cell culture[END_REF][START_REF] Helary | Dense fibrillar collagen matrices: A model to study myofibroblast behavior during wound healing[END_REF][START_REF] Vigier | Dense fibrillar collagen matrices and sponges for bone repair. Comparative study of structure and long-term culture of a human osteoblast cell line[END_REF][START_REF] Flanagan | Neurite branching on deformable substrates[END_REF] are put in the presence of a 3D environment. Such configuration presents a strong potential for the study of longrange cell migration mechanisms, for instance in the context of metastasis formation or wound healing. However, these thick matrices require large amounts of purified ECM macromolecules and are often not amenable to in situ observations like confocal microscopy.

Inclusion of cells within 3D macromolecular hydrogels (collagen, alginate, matrigel…) [START_REF] Bell | Production of a tissue-like structure by contraction of collagen lattices by human fibroblasts of different proliferative potential in vitro[END_REF][START_REF] Banerjee | The influence of hydrogel modulus on the proliferation and differentiation of encapsulated neural stem cells[END_REF][START_REF] Brannvall | Enhanced neuronal differentiation in a three-dimensional collagen-hyaluronan matrix[END_REF][START_REF] Lee | Three-dimensional cell culture matrices: state of the art[END_REF] is also a powerful approach to investigate specific cellular behavior like the ability to contract support matrices, the response to stress, the migration potential, etc. However, this well-proven technique has drawbacks, among which the stress generated by the encapsulation of cells during formation of the gel, and the fact that all components composing the gel are immediately in direct contact with the cells. Two-dimensional systems providing the cells with an adaptable three-dimensional network of physical and biochemical cues mimicking their natural niche [START_REF] Owen | Design of three-dimensional biomimetic scaffolds[END_REF] are a promising alternative to existing cell culture devices.

Here, we propose a system combining advantages of 3D culture supports, prepared by selfassembly with limited amounts of ECM macromolecules and an adaptable composition in relation with specific cell types. We improved the usability of collagen matrices as cell culture environments, suited for direct observations by confocal microscopy, by controlling their thickness and structure, and by chemically grafting them onto traditional glass supports. We show that different neurotrophic growth factors (Nerve growth factor, NGF and Brain derived neurotrophic factor, BDNF) can be included in the gels -impregnated or chemically conjugated -and effectively stimulate the differentiation of specific neural cell lines. PC-12 [START_REF] Greene | Establishment of a noradrenergic clonal line of rat adrenal pheochromocytoma cells which respond to nerve growth factor[END_REF] and SH-SY5Y [START_REF] Biedler | Multiple Neurotransmitter Synthesis by Human Neuroblastoma Cell Lines and Clones[END_REF] lines both exhibit high differentiation rates, dependent upon the substrate and the actual amount of the corresponding growth factor accessible to the cells, released or covalently coupled to the gel. The possibility to elicit neurite outgrowth in a 3D context with diffusing or immobilized growth factor raises the possibility of designing simplified brain tissue model materials.

Materials and methods

Culture set-up preparation and optimization

-Collagen extraction and purification

Type I collagen was extracted and purified from young Wistar rat tails following a protocol previously described [START_REF] Gobeaux | Fibrillogenesis in dense collagen solutions: A physicochemical study[END_REF]. Briefly, tendons were excised and washed with phosphate-buffered saline (PBS) to remove cells and traces of blood. Tissues were then soaked in a 4 M NaCl solution, rinsed and solubilized in 500 mM acetic acid to lyse remaining cells and precipitate some of the high-molecular-weight proteins. The solution was clarified by filtration and centrifugation (21,000×g for 2 hours). Collagen was then precipitated in 600 mM NaCl, centrifuged (6000×g for 45 min at 10°C) and finally solubilized in 500 mM acetic acid. The solutions were dialyzed (1:10 vol.) at least twice against 500 mM acetic acid to equilibrate the solvent and centrifuged (21,000×g for 4h at 15°C) to remove remaining impurities or aggregate. A collagen solution at 45 mg/ml was prepared by reverse dialysis against 75 g/l polyethylene glycol (20-35 kDa, Fluka) dissolved in 500 mM acetic acid. Collagen concentration was determined from the acidic solution before fibrillogenesis by assessing the amount of hydroxyproline [START_REF] Bergman | Two Improved and Simplified Methods for the Spectrophotometric Determination of Hydroxyproline[END_REF].

-Grafting onto glass coverslips

Covalent immobilization of collagen molecules on the glass surface of coverslips was carried out by successive surface modifications with an aminosysilane and a dialdehyde (Figure 1). Round coverslips (18 mm diameter) were thoroughly washed with methanol and distilled water before drying overnight at 110°C. They were then immersed in 2% Aminopropyltriethoxysilane (APTES from Sigma chemicals) in acetone for 3 hours at room temperature, and rinsed in acetone. Next, the coverslips were heated at 150°C for 1 hour to promote silanization of the surface. Then, the silanized surfaces were immersed in 5% glutaraldehyde (GA from Sigma Chemicals) in PBS for 2 hours at room temperature followed by rinsing with water, and drying in sterile conditions. Activated surfaces were later covered overnight at room temperature with an excess of collagen solution at 0.5 mg/ml (diluted with PBS from a 5 mg/ml collagen acetic acid solution). Grafted surfaces were washed with acetic acid (0.5 M), to remove the excess of ungrafted collagen molecules followed by deionized (milli-Q) water. Finally, reduction of Schiff's bases was achieved by immersing the coverslips into 0.02 M sodium borohydride (Sigma) in PBS for 1 hour at room temperature, followed by rinsing with deionized water.

-Calibrated hydrogel scaffold formation

A specially designed teflon mould was used to cast 50 µm-thick layers of acidic collagen solution (c.a. 45 mg/mL) onto collagen-grafted glass coverslips (Figure 1). A teflon lid was fixed and the whole setup submitted to ammonia vapours in a closed vessel to induce collagen fibrillogenesis. After three days, the samples were rinsed by immersing the mould for 12 hours in 0.25X PBS and freeze-dryed. The samples were stored in a dried atmosphere and all manipulations were done in sterile conditions.

-Growth factor impregnation and release

Lyophilised dense collagen matrices were hydrated overnight in a wet atmosphere with 50 µl of Dulbecco's Modified Eagle Medium (DMEM, Gibco®) containing 500 ng of NGF (Nerve Growth Factor 2.5S, Murine, Promega). The excess of liquid was then removed and the scaffolds were rinsed in PBS. Growth factor release kinetics were determined by immersing the gels in 1 ml of DMEM supplemented with 0.1% BSA and 0.02% sodium azide, in 12-well plates under orbital shaking (200 rpm) at room temperature. At exponentially spaced time points, small volumes (10 or 15 µL) of supernatant were collected in triplicate and replaced by fresh medium. The amount of NGF released was quantified by a sandwich ELISA (NGF E max ® ImmunoAssay System, Promega). Briefly, ELISA plates (Nunc-Immuno T M plates MaxiSorp T M , Fisher scientific) were coated with anti-NGF polyclonal antibody and blocked with the buffer provided. Soluble NGF from the sample is captured by the polyclonal antibody and further recognised by a monoclonal antibody. Finally, an HRP conjugated antiimmunoglobulin was added to the wells and after rinsing, the plates were incubated with TMB substrate and colour changes measured at 450 nm.

BDNF impregnation and release were performed in the same way. The medium was DMEM-F12 supplemented with 3% of Fetal Bovine Serum. BDNF (500 ng) was impregnated overnight at 4°C and the release was monitored at ambient temperature. Collected samples were kept at -20°C before quantification by sandwich ELISA (BDNF E max ® ImmunoAssay System, Promega).

-Growth factor coupling

Coupling of growth factors (GF), NGF and BDNF (recombinant human, Invitrogen) was performed on lyophilised gels first rehydrated with sterile PBS at pH 6. Activation was achieved by incubating the gels for approximately 8 hours at ambient temperature in a 33 mM EDC (1-Ethyl-3-(3-dimethylaminopropyl)carbodiimide, Sigma) and 66 mM NHS (Nhydroxysuccinimide, Sigma) solution in PBS at pH 6. After rinsing at least three times (10 min. in 1 ml of PBS, pH 7), collagen gels were kept immersed overnight at ambient temperature under orbital shaking at 200 rpm, in 1 ml of PBS at pH 7 containing 50 ng, 500 ng or no (control) growth factor. The medium was eliminated and the GF-coupled collagen gels were rinsed four times for at least 4 hours with 1 ml PBS pH 7 before plating the cells.

For colocalization experiments to evaluate the efficiency of the EDC/NHS coupling strategy, the coupling reaction was performed with collagen in solution. One ml of a 2 mg/ml collagen solution in MES buffer (pH 4.5) was mixed with 1 ml of 0.1M EDC-0.2M NHS solution in MES buffer and incubated with shaking for 30 minutes. The reaction was stopped with 0.4 M 2-mercaptoethanol for 15 minutes under shaking before adding NGF solution (shaking overnight at ambient temperature). Dialysis (Cellu1Sep® membrane, cut-off 6 kDa) was performed against PBS pH 7.4 for 4 hours to remove the remaining reagents and adjust the mixture to neutral pH to complete the reaction with NGF. Non-conjugated NGF was then removed by a second dialysis (Spectra/Por® Biotech membranes, cut-off: 50 kDa) and two successive centrifugations (30 min., 3,000×g). The pellet containing NGF-conjugated collagen fibres was placed on a microscopy slide and incubated with an anti-NGF monoclonal antibody (Promega) followed by an AlexaFluor® 594 (Invitrogen) secondary antibody for detection. Fluorescence and Differential Interference Contrast pictures were acquired on a

Zeiss Axioimager microscope at the Montpellier RIO Imaging Facility.

Materials characte rization. -Polarization Modulation -InfraRed Reflexion Absorption Spectroscopy (PM-IRRAS)

Experiments were carried out on 1 millimeter-thick grafted glass slides, in order to avoid interferences due to the low thickness of coverslips. PM-IRRAS spectra were recorded on a Nicolet 870 Fourier transform infrared (FT-IR) spectrometer by co-addition of 600 scans at a resolution of 8 cm -1 . The details of the optical setup, the experimental procedure, and the twochannel processing of the detected intensity have been described elsewhere [START_REF] Buffeteau | Polarization Modulation FT-IR Spectroscopy of surfaces and ultra-thin films -Experimental procedure and quantitative analysis[END_REF][START_REF] Blaudez | Investigation at the air/water interface using polarization modulation IR spectroscopy[END_REF]. All the spectra presented in this paper result from the ratio of the sample spectra to the substrate spectrum.

-Contact angle

Water contact angle measurements were performed using a Digidrop GBX fast/60 apparatus.

-Breaking strength

The efficiency of collagen grafting was estimated by performing ultimate tensile strength measurements on dense collagen matrices sandwiched between two grafted coverslips with a texture analyser apparatus TA.XT2 (Stable Micro Systems, Ltd in Godalming, Surrey UK) operated in the traction mode. The collagen solution (45 mg/ml in acetic acid) was deposited with a thickness of about 1 mm defined by glass spacers between two grafted coverslips and was submitted to ammonia vapours overnight to induce fibrillogenesis. The samples were attached to the static and moving parts of the apparatus, by means of metal connectors glued to the free glass surfaces on each side. Tensile tests were performed with a constant velocity of 0.1 mm/s, and the ultimate breaking strength was determined.

-Scanning Electron Microscopy (SEM)

For SEM observations, collagen scaffolds were previously fixed by immersion in a glutaraldehyde solution (2.5% in PBS) for 1 hour at ambient temperature and dehydrated in water/ethanol mixtures (10%, 30%, 50%, 70% and 100% ethanol). Samples were then dried by supercritical drying with CO 2 (E3100 Polaron Critical Point drier). After platinum-sputter coating (Balzers SCD 020) samples were observed in the secondary electron mode with a scanning electron microscope (Hitachi 4800 S) operated at an accelerating voltage of 50 kV.

In order to prevent gel collapse and measure gels thickness, edges of samples were observed under low vacuum (0.7 torr) with a Quanta 200F FEI microscope at an accelerating voltage of 15 kV.

-Growth factor distribution through the gel

To characterize the spatial distribution of the coupled growth factor (NGF), gels were fixed at room temperature 1 hour with 4% paraformaldehyde (PFA) and incubated in 30% sucrose overnight. The samples were then placed face down on moulds filled with Tissue-Tek® O.C.T. TM compound (Sakura*, Finetek), frozen in liquid nitrogen and stored at -20°C before cryosectionning. Cryosections, 20 µm-thick, were obtained perpendicular to the gels, with a 2800N Jung Frigocut cryostat (Leika). The slides were air-dried at room temperature in a humidified chamber for at least 30 minutes and then blocked in PBS, 2% BSA, 0.05% Triton.

Samples were incubated with anti-NGF monoclonal antibody (Promega, diluted 1/1000) overnight at 4°C. After extensive rinsing, the slides were incubated in the secondary antibody AlexaFluor® 594 (Invitrogen) and subsequently mounted with Prolong® Gold antifade reagent (Invitrogen). Stained collagen gel sections were observed on Zeiss Axioimager. To generate fluorescence intensity profiles, images were acquired on a DM 2500 Leica confocal microscope on three different sections. Four intensity profiles were generated with the ImageJ (National Institute of Health) freeware for each section at different places and distances were normalized by the total thickness of the gel section.

-Cell culture and differentiation

PC-12 cells. The rat pheochromocytoma cell line was obtained from the American Type Culture Collection (ATCC, Rockville, MD, USA). PC-12 cells were maintained in Dulbecco's Modified Eagle Medium (DMEM, Gibco®) supplemented with 2 mM glutamine, 5% heatinactivated fetal bovine serum, 10% Horse serum and 1% antibiotics (penicillin, streptomycin), in a humidified incubator, under an atmosphere of 5% CO 2 , at 37°C. For differentiation, PC-12 cells were seeded in DMEM without serum in 12-well plates at a density of 50,000 cells per well. Substrates for PC-12 cells differentiation were dense collagen matrices prepared as described above, or conventional collagen-coated coverslips as controls.

Differentiation controls consist in adding 50 ng of NGF to the medium. NGF-releasing gels were prepared the day before seeding the cells, by impregnating the collagen scaffolds with 50 µl of a NGF solution at 10 µg/ml in DMEM overnight in a humid atmosphere. NGFcoupled collagen gels were used immediately after rinsing.

SH-SY5Y cells. SH-SY5Y neuroblastoma line was purchased from ATCC and routinely grown in DMEM-F12 with 2 mM glutamine, 10% fetal bovine serum, 1% antibiotics (penicillin, streptomycin), and fungizone® antimycotic (Gibco®, 2.5 µg/ml). SH-SY5Y cells were seeded on collagen substrates in 12-well plates at a density of 100,000 cells per well and maintained 24 hours in DMEM-F12 with 10% fetal bovine serum. SH-SY5Y cells were then differentiated in DMEM-F12 supplemented with 3% fetal bovine serum and 6.8 µM all-trans retinoic acid (Sigma). BDNF-releasing gels were prepared the day before seeding the cells, by impregnating the collagen scaffolds with 50 µl of a BDNF solution at 1 µg/ml in DMEM-F12.

BDNF-coupled collagen gels were used immediately after rinsing and compared to differentiation controls with 50 ng BDNF (Recombinant human BDNF, Gibco®).

Cells immunofluorescent labelling and imaging

Cells were fixed in paraformaldehyde 4% (Sigma) in PBS for 30 minutes after 4 days of differentiation for PC-12 cells and 6 days for SH-SY5Y cells. After fixation and washing in PBS, cells were blocked and permeabilized in 2% BSA and 0.1% triton X-100 in PBS for 20 minutes. Samples were incubated with primary antibody monoclonal anti-1III-tubulin (mouse monoclonal, 1/200, Sigma) overnight at 4°C. After three washes in PBS, samples were incubated with the secondary antibody Cy3 Donkey anti-mouse IgG (1/400, Jackson

ImmunoResearch Laboratories) for one hour at room temperature. During the last wash, nuclei were counterstained by incubating in DAPI (300 mM, Invitrogen) for 5 minutes before mounting with ProLong® Gold Antifade reagent (Invitrogen).

Fluorescence images were acquired on DM 2500 Leica confocal microscope. Images stacks were reconstructed with ImageJ.

Neurite outgrowth measure ment

Neurites outgrowth was quantified on epifluorescence images (Zeiss Axioimager). Images of 5 fields per well were taken. Neurite growth was determined by manually tracing the length of (1) all the neurites which could be associated with a particular cell body and (2) the longest neurite per cell for all cells in a field that had an identifiable neurite and for which the entire neurite arbor could be visualized. We measured the length and total output (sum length) of only primary neurites. Neuronal processes were analyzed using the Axiovision LE 4.8 software. For each graph, neurite length data were generated from at least three independent experiments and more than 80 cells were counted for each experiment. Data were analyzed by Student's t test. We consider that a cell is differentiated when it possesses at least one neurite length superior or equal to two fold the cell body diameter.

Results and discussion

-Collagen biomimetic matrices -Grafting onto glass surfaces

Our experience told us that relatively thick (50 µm) collagen hydrogels do not strongly adhere to glass and often detach from the coated coverslips during manipulation (gel preparation, culture, staining, mounting…), which can seriously impair cellular investigations. In what follows, we characterize the anchoring of dense fibrillar collagen matrices to glass surfaces, as schematically depicted in figure 1 through a combination of chemical coupling and physical gel formation. The first part of the procedure consists in covalently attaching collagen molecules to the glass surface. Contact angle measurements were performed after each step to determine the evolution of the surface energy as a first indicator of the grafting efficiency.

Bare glass slides, ie without any modifications, have a mean water contact angle of 56±1°.

After silanization with APTES and then coupling of glutaraldehyde, the contact angles were respectively 78±2° and 74±6°, indicating increased hydrophobicity attributed to the hydrocarbon chains of APTES and GA. After collagen immobilization, the water contact angle decreased back to 45±6°, consistent with the high hydrophilicity of most proteins.

Grafting steps were followed at the molecular scale by PM-IRRAS. The silanized and glutaraldehyde-treated surfaces (Figures 2a and2b) displayed a large positive band between 1460 and 1440 cm -1 attributed to CH 2 bending, and a negative band at 1581 cm -1 assigned to NH 2 bending. The negative sign of the last band indicates a specific orientation of the transition moment, perpendicular to the substrate surface. After addition of collagen and extensive washing (Figure 2c), the characteristic amide I (~1650 cm -1 ) and amide II (~1550 cm -1 ) bands of proteins are observed. Unlike for bulk collagen, where the amide I band is much stronger than the amide II [START_REF] Doyle | Infrared Spectroscopy of Collagen and Collagen-Like Polypeptides[END_REF], in the present case of coupled triple helices, both bands display nearly the same intensities. This unusual ratio indicates an anisotropic orientation of collagen molecules relative to the substrate. In the collagen triple helix structure [START_REF] Brodsky | The collagen triple-helix structure[END_REF], C=O and NH links are roughly perpendicular to the helix axis, resulting in respectively parallel and perpendicular amide II and amide I transitions moments. Considering surface selection rules of PM-IRRAS, the low amide I/amide II intensity ratio observed in our spectra can thus be attributed to a globally flat orientation of triple helices on the substrate in the dry state.

After a first monolayer of collagen was covalently attached to the glass surface, a thick gel was cast in a mould of defined thickness by pouring an acidic collagen solution at a chosen concentration (here 45 mg/mL) then submitted to ammonia vapors to induce fibrillogenesis [START_REF] Gobeaux | Fibrillogenesis in dense collagen solutions: A physicochemical study[END_REF][START_REF] Besseau | Production of ordered collagen matrices for three-dimensional cell culture[END_REF]. Ultimately, the gel grafting efficiency was assessed qualitatively by the sustained adhesion of the collagen matrices to the glass supports throughout the sample preparation steps and cell culture experiments. In an attempt at quantifying the functional gain of grafting the gels, we estimated the force necessary to detach them from the glass coverslip by measuring the tensile breaking strength for grafted samples compared to ungrafted ones. This was done by pulling on thick gels sandwiched between two coverslips, in the presence or absence of chemical grafting. In all cases, the rupture appeared to occur at the interface between the coverslip and the gel and not in the bulk of the gel. The breaking strength was

shown to increase about six times, from approximately 5±0.2 kN.m -² for ungrafted samples, to about 26±4.4 kN.m -² for grafted ones. Ultimate tensile strength measured by other groups for similar collagen materials prepared in vitro being rather on the order of the MPa [START_REF] Christiansen | Assembly of type I collagen: fusion of fibril subunits and the influence of fibril diameter on mechanical properties[END_REF][START_REF] Sheu | Characterization of collagen gel solutions and collagen matrices for cell culture[END_REF], the interface appears weaker than the bulk fibrillar gel. This is probably because the density of physical reticulation nodes (native collagen fibrils) within the gel is much higher than the density of covalent bonds between the gel and the glass surface, resulting in a lower mechanical strength at the interface. In any case, the simple grafting process presented here proved sufficient to prevent detachment of the gels during the following experimental procedures.

-Biomimetic structure and 3D organization Native collagen fibrils, with the typical 67 nm -periodic striation, are visible by SEM (Figure 3) on the surface of the dry collagen scaffolds. The fibrils have diameters ca. 100 nm and appear locally aligned. The samples thickness was estimated in SEM (edge view) under low vacuum conditions slightly below 40 µm after supercritical CO 2 drying. The difference with the height defined by the mould geometry (50 µm) can be attributed to the contraction of samples during fibrillogenesis and/or a partial collapse of the gel during supercritical CO 2 drying. However, there is no significant difference neither in structure nor in thickness, between gels supercritically dried in CO 2 immediately after chemical fixation, and gels previously freeze-dried and rehydrated (corresponding to the growth factor impregnation step)

prior to supercritical drying. Overall, the gels appear dense but porous with a flat surface exposed to the cells to be seeded on top.

-PC-12 cells behavior on NGF-containing collagen hydrogels

Although type I collagen is a common substrate for the culture and neural differentiation of PC-12 cells [START_REF] Greene | Establishment of a noradrenergic clonal line of rat adrenal pheochromocytoma cells which respond to nerve growth factor[END_REF][START_REF] Fujii | Neurite outgrowth and protein synthesis by PC12 cells as a function of substratum and nerve growth factor[END_REF], we first assessed that they indeed survive and differentiate on these relatively soft hydrogels, at least compared to coated glass or plastic. When 50 ng NGF are added to the culture medium without serum, the proportion of cells showing the morphological features of differentiated cells after 4 days reaches about 70% (data not shown), close to the maximum value reported previously [START_REF] Fujii | Neurite outgrowth and protein synthesis by PC12 cells as a function of substratum and nerve growth factor[END_REF] with conventional coatings.

When NGF was included in the gel by simple impregnation, PC-12 cells also exhibited a neural phenotype with numerous elongated cellular processes and branching, as illustrated by 1-3 tubulin staining in Figure 4A after four days of culture in serum-free medium. It is noteworthy that PC-12 cultured in the culture medium without NGF and in the absence of serum do not differentiate at all and most often die within a few days [START_REF] Greene | Nerve growth factor prevents the death and stimulates the neuronal differentiation of clonal PC12 pheochromocytoma cells in serum-free medium[END_REF]. Therefore, as described previously for other hydrogels [START_REF] Beaty | Controlled growth factor delivery induces differential neurite outgrowth in three-dimensional cell cultures[END_REF][START_REF] Willerth | Rationally designed peptides for controlled release of nerve growth factor from fibrin matrices[END_REF][START_REF] Doleski | NGF release from thermo-responsive collagen-polyNIPAam polymer networks supports neuronal cell growth and differentiation[END_REF] or thin collagen coatings [START_REF] Bhang | The effect of the controlled release of nerve growth factor from collagen gel on the efficiency of neural cell culture[END_REF], active NGF is released from the thick matrix and stimulates neurite outgrowth. However, to reach a similar differentiation level (Figure 4B,C, t-test analysis), larger amounts (500 ng) of impregnated NGF were necessary compared to the condition where the growth factor is directly added to the medium (50 ng).

This difference can be in part ascribed to the release kinetic of NGF from the gel and its delivery to the cells. The quantification by ELISA of the time-dependant amount of native NGF released in the medium (Figure 5A) indicates that only ~1/3 (35%) of the initial impregnated amount is released and detectable by ELISA over the course of our experiments (4 days). This suggests that a significant proportion of the impregnated growth factor could be retained in the gel. Although to our knowledge, no specific interaction sequence has been identified, adsorption of NGF on collagen coatings has been reported before [START_REF] Sun | Collagen scaffolds loaded with collagen-binding NGF-beta accelerate ulcer healing[END_REF][START_REF] Ahmad | Endostatin binds nerve growth factor and thereby inhibits neurite outgrowth and neuronal migration in-vitro[END_REF]. The spontaneous deactivation of NGF, especially in dilute solutions [START_REF] Nguyen | Stability and Interactions of Recombinant Human Nerve Growth Factor in Different Biological Matrices: In Vitro and In Vivo Studies[END_REF] could also explain most of the difference between the loaded amount and the quantity measured in the medium after four days of release from the gel (personal data). Considering therefore that only part of the growth factor is accessible to the cells, and given the strong NGF dose-response dependence in the 0-50 ng range reported for PC-12 [START_REF] Greene | Nerve growth factor prevents the death and stimulates the neuronal differentiation of clonal PC12 pheochromocytoma cells in serum-free medium[END_REF][START_REF] Katzir | A quantitative bioassay for nerve growth factor, using PC12 clones expressing different levels of trkA receptors[END_REF][START_REF] Schimmelpfeng | Quantification of NGF-dependent neuronal differentiation of PC-12 cells by means of neurofilament-L mRNA expression and neuronal outgrowth[END_REF], the amount released with 50 ng of NGF impregnated in the gel (~10-15 ng) would be insufficient to obtain high differentiation levels.

Therefore, to ensure optimum differentiation in all conditions, we chose to work in the present study, with an excess of immobilized growth factor (500 ng), which could probably be reduced by at least a factor of 3 and still induce a neural phenotype for a majority of PC-12 cells.

One major advantage of using thick dense collagen matrices to differentiate neural cells is that neurite outgrowth is in principle possible in 3D within the matrix supporting the cells. In these conditions we were able to observe, by confocal microscopy, differentiated cells with neurites growing several tens of microns into the depth of the gels (Figure 5B). Here, the growth factor necessary to the cells differentiation is delivered from the gel. It is absent from the medium when the cells are seeded, which results in a steep concentration difference between the interior and exterior of the gel. The slow release of NGF from the matrix should result in the formation of concentration gradients around the gel/medium interface [START_REF] Willerth | Rationally designed peptides for controlled release of nerve growth factor from fibrin matrices[END_REF], the bottom of the gel acting as a reservoir. However, such gradient is transitory and disappears once the system approaches equilibrium (plateau in the release kinetic), which occurs in about 24 hours. The chemotactic effect [START_REF] Cao | Defining the concentration gradient of nerve growth factor for guided neurite outgrowth[END_REF][START_REF] Beaty | Controlled growth factor delivery induces differential neurite outgrowth in three-dimensional cell cultures[END_REF] expected from the establishment of diffusion-induced NGF concentration gradients appears too limited [START_REF] Kapur | Immobilized concentration gradients of nerve growth factor guide neurite outgrowth[END_REF] to guide massively the growth of neurites into the gel.

-Efficiency of growth factor covalently coupled to collagen matrices

We thus turned to the covalent coupling of the growth factor to enhance the chemotactic effect of the NGF included in the gel and foster 3D neuritic growth. The EDC/NHS [START_REF] Shen | Vascular endothelial growth factor immobilized in collagen scaffold promotes penetration and proliferation of endothelial cells[END_REF][START_REF] Wissink | Immobilization of heparin to EDC/NHS-crosslinked collagen. Characterization and in vitro evaluation[END_REF] coupling efficiency was evaluated by immuno-fluorescent staining of NGF on cryosections of collagen gels (Figure 6A-D). An exposure time of 1.4s is required to reveal the control gels without NGF (Figure 6A) or without linkers (Figure 6B), while NGF-coupled gels are overexposed (Figure 6C). The much larger fluorescence intensity observed for the former indicates efficient coupling, as confirmed by the colocalization of immuno-fluorescently labeled NGF with large isolated collagen fibrils observed in the differential interference contrast (DIC) mode (Figure 6F). Concentration profiles of collagen-coupled NGF were established by measuring the fluorescence intensity across the gel, ie along the direction perpendicular to the surface (Figure 6D-E). The density of coupled NGF increases moderately and gradually throughout the gel from the glass coverslip to the scaffold surface. Such gradient is not unexpected considering the simple one-way diffusion of NGF from the top of the gel to the bottom and its subsequent coupling to the NHS-activated collagen.

NGF-coupled collagen gels proved efficient to induce a neural phenotype in PC-12 cells.

With 50 ng of NGF coupled to the gel (Figure 7A, 50 ng -c-gel) the differentiation rate is slightly increased with respect to control (Figure 7A, No linkers), but remains much lower (31.6%) than when the same amount is added in a soluble form to the culture medium (70.2%) (Figure 7A, 50 ng @ med)). Adding 500 ng of NGF to the matrix (Figure 7A, 500 ng -c-gel) restores a high level of differentiation (61.1%) with well developed neurites extending over several times the size of the cell body as illustrated in Figure 7B. The total length of neurites emitted by individual differentiated cells on 50 and 500 ng NGF-coupled collagen gels is also significantly increased compared to the control without linkers (Figure 7C). This clearly indicates that at least part of the bound growth factor is active. It should be noted that we did not prime the cells by exposing them to soluble NGF before seeding them on the matrices. Therefore, the neural phenotype was principally initiated by growth factor molecules coupled near the surface. As reported previously for NGF [START_REF] Kapur | Immobilized concentration gradients of nerve growth factor guide neurite outgrowth[END_REF][START_REF] Yu | The use of immobilized neurotrophins to support neuron survival and guide nerve fiber growth in compartmentalized chambers[END_REF], other growth factors or proteins [START_REF] Shen | Vascular endothelial growth factor immobilized in collagen scaffold promotes penetration and proliferation of endothelial cells[END_REF][START_REF] Pohl | Surface immobilization of bone morphogenetic protein 2 via a self-assembled monolayer formation induces cell differentiation[END_REF][START_REF] Minato | Cardiac differentiation of embryonic stem cells by substrate immobilization of insulin-like growth factor binding protein 4 with elastin-like polypeptides[END_REF], interaction of the active molecule covalently coupled to the matrix with corresponding membrane receptors is enough to induce cell survival and in some cases differentiation.

Yu et al. [START_REF] Yu | The use of immobilized neurotrophins to support neuron survival and guide nerve fiber growth in compartmentalized chambers[END_REF] evaluated that a 30 ng/cm 2 surface density of bound NGF yields a similar neural survival rate (~75%) as with 50 ng/ml soluble NGF, while a 10-fold reduction to 3 ng/cm 2 brought the survival rate down to less than 40%. In the present study, the growth factor is coupled to the whole gel and only the surface is initially accessible to cells. Given the rugosity of the dense collagen matrix surface and considering, for the sake of rough estimation, that about one tenth (5 µm-deep) of NGF is directly accessible yields NGF surface densities of about 2 ng/cm 2 (50 ng coupled) and 20 ng/cm 2 (500 ng coupled). Although the NGF doseresponse of PC-12 survival and differentiation rates are probably different, this dependence might explain the slightly lower differentiation rate observed with 500 ng coupled NGF as compared to its addition at 50 ng/ml in the medium.

We also observed neurites growing in the depth of the gel (Figure 7D). Although this might not be considered strictly as "neurite guidance" [START_REF] Cao | Defining the concentration gradient of nerve growth factor for guided neurite outgrowth[END_REF][START_REF] Kapur | Immobilized concentration gradients of nerve growth factor guide neurite outgrowth[END_REF][START_REF] Cao | Investigating the synergistic effect of combined neurotrophic factor concentration gradients to guide axonal growth[END_REF][START_REF] Mai | Axon initiation and growth cone turning on bound protein gradients[END_REF], the topological and biochemical cues of native collagen fibrils combined to the presence of coupled growth factors are therefore enough for neurites to grow within the gel.

-Neuroblastoma cells culture and differentiation

To demonstrate that our cell differentiation support is amenable to a variety of neural cells, we investigated the behavior of human neuroblastoma cells (SH-SY5Y). This cell line, derived from tumors of the human peripheral neural system, was chosen because it presents many of the characteristics of primary cultures of neurons and was often used to study molecular and cellular features of neurodegenerative diseases like AD [START_REF] Paris | Anatabine lowers Alzheimer's Abeta production in vitro and in vivo[END_REF][START_REF] Wang | Zinc Overload Enhances APP Cleavage and A1 Deposition in the Alzheimer Mouse Brain[END_REF]. When seeded on collagen hydrogels in the presence of 6.8 µM retinoic acid (RA), about 30% of the SH-SY5Y cells are differentiated, even in the absence of BDNF (Figure 8A and B, 0 ng BDNF).

This proportion rises to more than 60% upon addition of 50 ng of BDNF in the medium (Figure 8A and B, 50 ng @ med). The same amount impregnated in the collagen matrix (Figure 8A and B, 50 ng @ gel) also induces a significative increase of differentiated cells compared to the control condition treated with RA only. Neurite length is slightly increased when BDNF is added either in the medium or in the gel by impregnation (Figure 8C).

Surprisingly, the BDNF release profile determined by ELISA (Figure 8D) shows a maximum at 24 hours (100 ng; 20% of the loaded amount), followed by a slow decrease. We determined that the proportion of native (ELISA-detectable) BDNF decreases exponentially with time with an exponent value (~e -kt , k = 0.04 day -1 ) very close to that found previously by Callahan et al. (0.03 day -1 ) [START_REF] Callahan | Sodium Chloride Enhances the Storage and Conformational Stability of BDNF and PEG-BDNF[END_REF]. It also appears that BDNF largely adsorbed onto the polystyrene plastic walls of the culture plate despite the presence of BSA-containing serum in the release medium. This strong adsorption, which does not occur in polypropylene microtubes (data not shown), amounts to about 300 ng, ie ~60% of the initial amount in solution after 24 hours.

The combination of spontaneous BDNF degradation and its adsorption to the plate well surface explain the non-monotonic release profile in figure 8D.

BDNF chemically coupled (50 ng) to the gels is also able to stimulate the differentiation of SH-SY5Y as illustrated in figure 8A. The differentiation rate (Figure 8B) and neurite length (Figure 8C) are slightly increased compared to the control with RA only and are similar to those obtained with the same amount of soluble growth factor. Treatment of SH-SY5Y with RA at concentrations in the µM range is known to potentiate their responsiveness to the subsequent addition of BDNF [START_REF] Encinas | Extracellular-Regulated Kinases and Phosphatidylinositol 3-Kinase Are Involved in Brain-Derived Neurotrophic Factor-Mediated Survival and neuritogenesis of the Neuroblastoma Cell Line SH-SY5Y[END_REF][START_REF] Encinas | Sequential Treatment of SH-SY5Y Cells with Retinoic Acid and Brain-Derived Neurotrophic Factor Gives Rise to Fully Differentiated, Neurotrophic Factor-Dependent, Human Neuron-Like Cells[END_REF][START_REF] Matsumoto | Expression of Brainderived Neurotrophic Factor and p145TrkB Affects Survival, Differentiation, and Invasiveness of Human Neuroblastoma Cells[END_REF][START_REF] Kaplan | Induction of TrkB by retinoic acid mediates biologic responsiveness to BDNF and differentiation of human neuroblastoma cells[END_REF]. It is therefore expected that, provided enough growth factor is available for the cells, typically 50-100 ng/ml [START_REF] Encinas | Extracellular-Regulated Kinases and Phosphatidylinositol 3-Kinase Are Involved in Brain-Derived Neurotrophic Factor-Mediated Survival and neuritogenesis of the Neuroblastoma Cell Line SH-SY5Y[END_REF][START_REF] Encinas | Sequential Treatment of SH-SY5Y Cells with Retinoic Acid and Brain-Derived Neurotrophic Factor Gives Rise to Fully Differentiated, Neurotrophic Factor-Dependent, Human Neuron-Like Cells[END_REF][START_REF] Matsumoto | Expression of Brainderived Neurotrophic Factor and p145TrkB Affects Survival, Differentiation, and Invasiveness of Human Neuroblastoma Cells[END_REF], it induces the production of numerous long processes indicative of neural differentiation. These data again indicate that the EDC/NHS covalent coupling method enables the interaction of the growth factor with its membrane receptors and activation of the appropriate cellular signalization pathways.

Interestingly, in all experiments we mostly observed N-type cells and very few S-type cells, although the cells were kept in the presence of RA and serum (3% Horse Serum) during their treatment with BDNF [START_REF] Encinas | Sequential Treatment of SH-SY5Y Cells with Retinoic Acid and Brain-Derived Neurotrophic Factor Gives Rise to Fully Differentiated, Neurotrophic Factor-Dependent, Human Neuron-Like Cells[END_REF][START_REF] Påhlman | Retinoic acidinduced differentiation of cultured human neuroblastoma cells: a comparison with phorbolester-induced differentiation[END_REF].

Conclusions

We functionalized calibrated dense fibrillar collagen matrices to differentiate cell types widely used as models to study neural behavior. We successfully cultured and induced PC-12 and SH-SY5Y cells into a neural phenotype on thick (ca. 50 µm) collagen hydrogels with appropriate growth factors, respectively NGF and BDNF. When impregnated in dense matrices and delivered to the cells by simple diffusion-limited release, the trophic effect of both neurotrophins was enough to reach high proportions of differentiated cells (about 45 to 65 %). In addition, both NGF and BDNF chemically conjugated to the collagen matrices were also able to induce a neural phenotype, suggesting that intracellular signaling was efficiently triggered by their interaction with membrane receptors. For both impregnated and conjugated GF, we observed neurites growing several tens of microns into the depth of the fibrillar gels.

The scaffolds described herein were prepared by simple self-assembly processes with readily adaptable structure (porosity, collagen density, fibril size…) and controlled geometry. They are particularly suitable for the study of neurite growth in a biomimetic reconstructed matrix and direct observation by confocal microscopy. The simple and versatile coupling chemistry and preparation protocols make it easy to elaborate 3D cell microenvironments adapted for cellular investigations, especially those requiring vertical compartmentation. For instance, it should be a valuable tool to study the relationship between Alzheimer's diseases and neurite dystrophy, which has been shown to be at least two-prone. First, dystrophic neurites are associated to degenerating neurons and to amyloid 1-peptide (A1) aggregates in senile plaques [START_REF] Knowles | Plaqueinduced neurite abnormalities: Implications for disruption of neural networks in Alzheimer's disease[END_REF][START_REF] Dickson | The morphological phenotype of 1-amyloid plaques and associated neuritic changes in Alzheimer's disease[END_REF][START_REF] Duyckaerts | Classification and basic pathology of Alzheimer disease[END_REF]. Second, it was shown in previous works using compartmented systems that neurites isolated from the cell body treated with A1 aggregates undergo degeneration followed by cell death [START_REF] Song | Apoptosis is secondary to non-apoptotic axonal degeneration in neurons exposed to Aβ in distal axons[END_REF]. However, investigating the specific role of such neurite / A1 interactions in neuron death with 2D culture systems is hampered by the difficulty to truly treat the neurites separately from the cell body, while permitting their growth from differentiating cells. To test this hypothesis, we are currently investigating the adverse effects of Aβ-amyloid aggregates included in a 3D biomimetic collagen matrix, in the vicinity of growing neurites.

Figure 1 : 2 )

 12 Figure 1: Collagen grafting and thick dense matrices elaboration. (1) Conventional glass coverslips were functionalized by silanization with APTES and glutaraldehyde before covalent grafting of collagen triple helices. (2) An acidic concentrated collagen solution was cast on the top of the grafted coverslip to form, after (3) exposition to NH 3 vapors, a 50 µmthick fibrillar collagen gel (4).

Figure 2 :

 2 Figure 2: PM-IRRAS spectra of the glass coverslip surface after (a) silanization with APTES, (b) glutaraldehyde reaction and, (c) collagen triples helix coupling.

Figure 3 :

 3 Figure 3: Representative SEM images of fibrillar collagen scaffolds after chemical fixation and supercritical drying. A and B: Samples kept hydrated after fibrillogenesis under ammonia vapors. C and D: Samples lyophilized after gel formation and then impregnated with phosphate buffer. A and C: Surface pictures, scale bar 1 µm. B and D: Edge view visualized under low vacuum, scale bar 40 µm.

Figure 4 :

 4 Figure 4: Effect of NGF added to the medium or impregnated within gels on PC-12 cells differentiation without serum. 50 ng of NGF were added to the medium (50 ng @ med) or 500 ng included in the gel by impregnation (500 ng @ gel). A: Immunofluorescence analysis of differentiated PC-12 cells after 1-3 tubulin (red) and DAPI (blue) staining. Images were acquired in confocal microscopy (Z-projections of 4 confocal optical sections, interval 0.5 µm). Scale bar 20 µm. B: Quantitative analysis of undifferentiated and differentiated PC-12 cells. C: Total neurite length per cell was measured and expressed as mean ± SEM.

Figure 5 :

 5 Figure 5: A: Release profile of NGF from fibrillar collagen gels measured by ELISA. The quantity (ng) and proportion of the initial amount expressed as mean ± SD were obtained with three distinct experiments with sampling performed in triplicate. B: Immunofluorescence images of differentiated PC-12 cells after 1-3 tubulin (red) and DAPI (blue) staining. Images were acquired in confocal microscopy (49 confocal optical sections, interval 0.5 µm). Top, Zprojection of confocal slices; bottom, side-view reconstruction of the 49 confocal sections showing neurite growth through the gel. Scale bar 20 µm.

Figure 6 :

 6 Figure 6: Spatial distribution and efficiency of NGF coupling to collagen gels. A-D: Immunofluorescence images of cryosectionned dense collagen matrices stained with an anti-NGF antibody: control gel without NGF (A), control gel without crosslinkers (B) and NGFcoupled collagen gel (C-D). The images were obtained with the same laser excitation and an exposure time of 1.4 s (A-C) or 0.3 s (D). Scale bars 50 µm. E: Fluorescent intensity profile of NGF coupled to collagen as a function of the normalized distance from the coverslip surface (Grey curves: ± 12). F: Co-localization of the anti-NGF immunofluorescent labeling and collagen fibers observed in the DIC mode. Scale bar 5 µm.

Figure 7 :

 7 Figure 7: Effect of collagen coupled NGF on PC-12 cells differentiation. A: Quantitative analysis of differentiated and undifferentiated PC-12 cells. B: Immunofluorescence images of differentiated PC-12 cells after 1-tubulin (red) and DAPI (blue) staining. Images were visualized by confocal microscopy (Z-projections of 4 confocal optical sections, interval 0.5 µm). Scale bar 20 µm. C: The total neurite length per cell was measured and expressed as mean ± SEM. Asterisks indicate significant differences from control without crosslinkers following student t-test analysis. (* p<0.05, ** p<0.01 *** p<0.001). D: Immunofluorescence reconstructed (49 confocal optical sections, interval 0.5 µm) side-view image of differentiated PC-12 cells after 1-tubulin staining. Scale bar 20 µm.

Figure 8 :

 8 Figure 8: Effect of BDNF in the medium, impregnated or covalently coupled into the gels on SH-SY5Y cells differentiation. SH-SY5Y cells were cultured with 6.8 µM retinoic acid. BDNF (50 ng) was added to the medium (@ med), included in the gel by impregnation (@ gel) or chemically conjugated (-c-gel). A: Immunofluorescence images of differentiated SH-SY5Y cells after 1-3 tubulin (red) and DAPI (blue) staining. Images were visualized by confocal microscopy (Z-projections of 5 confocal optical sections, interval 0.5 µm). Scale bar 50 µm. B: Quantitative analysis of differentiated and undifferentiated SH-SY5Y cells. Asterisks indicate significant differences from control without BDNF following student t-test analysis (* p<0.05). C: The total neurite length per cell was measured and expressed as mean ± SEM. D: Release profile of BDNF from fibrillar collagen gels measured by ELISA. The quantity (ng) and the proportion of the initial amount released into the medium (mean±SD) were obtained with three distinct experiments and sampling performed in triplicate.

FIGURES Figure 1 Figure 3 Figure 5 Figure 6 Figure 8

 13568 FIGURES

Acknowledgme nts

The authors are grateful to Daniel Brunel for fruitful discussions and advice regarding the coupling chemistry and to Bernard Desbat for the PM-IRRAS measurements. We wish to thank Vicky Diakou for her assistance at the Montpellier RIO Imaging facility.


Jean-Marc L'hermite
email: j-m.lhermite@irsamc.ups-tlse.fr

Sebastien Zamith

Macroscopic extension of RRK and Weisskopf models of unimolecular evaporation

Keywords: 36, 40, Qv -Stability and fragmentation of clusters 64, 70, fm -Thermodynamics studies of evaporation and condensation

The macroscopic behavior of two microscopic unimolecular evaporation models is examined. The temperature dependence of bulk equilibrium vapor pressures deduced from the Rice-Ramsperger-Kassel (RRK) model and from the Weisskopf model, respectively, are compared. The bulk vapor pressures of sodium given by both models are compared, over a wide temperature range, with the experimental vapor pressures. The Weisskopf model is in better agreement with the experimental data than the RRK theory: From the melting temperature to the critical temperature (370K-2500K), the vapor pressures of sodium calculated using the Weisskopf model agree with experimental values within 2%, whereas RRK theory leads to errors of more than 40%. Weisskopf theory satisfying detailed balance principle, whereas RRK theory does not, may explain this result.

Introduction

Evaporation is an ubiquitous physical phenomenon in which single particles escape from the solid or liquid state into the gas phase.

At the macroscopic scale, evaporated particles form a vapor, whose pressure depends on the nature of the substance and of its temperature. The evaporation rate of atoms or molecules from the bulk in solid or liquid state is characterized by its equilibrium vapor pressure, at which the evaporation probability exactly counterbalances the condensation probability. Many experimental methods exist since a long time for determining equilibrium vapor pressures of various substances [START_REF] Ditchburn | For monoatomic vapors[END_REF].

At the molecular scale, thermal evaporation of atoms or molecules from single small objects, such as molecules or clusters, cannot be described, nor experimentally characterized, by an equilibrium vapor pressure, which is a statistical description of a large number of particles. However, the evaporation rate of isolated small particles can be experimentally investigated by mass spectrometry methods [2].

Some phenomena, such as droplets nucleation in a supersaturated vapor for instance, require a description of evaporation that should be valid at both microscopic and macroscopic scales. One way to obtain such a description is to extend asymptotically to infinite size systems the expressions for evaporation and condensation given by microscopic models. The macroscopic evaporation rates obtained must of course obey the known bulk properties; in particular, they must be able to reproduce the experimental evolution of equilibrium vapor pressures with temperature.

The present paper aims at establishing a bridge between microscopic models of thermal evaporation and the related properties of bulk. Experimental vapor pressures of sodium are compared here to calculated values obtained from the extension to macroscopic systems of two models widely used to describe microscopic evaporation: the RRK model and the Weisskopf model. Sodium is chosen as a typical example since the evaporation of sodium clusters is well documented and both RRK and Weisskopf models are able to reproduce microscopic experimental data [2].

A significant difference between Weisskopf and RRK approaches, which has important consequences as it will be shown, is that the RRK model does not account for the density of states of the evaporated particle (it is a non-microreversible model), whereas the Weisskopf model, which obeys the detailed balance principle (it is a microreversible model), does. It will be shown that this issue with RRK theory, which was already pointed out [3,4,[START_REF] Lin | Proc. Nat. Acad. Sci[END_REF], makes it incompatible with classical thermodynamics when applied to macroscopic systems.

The simple analytical expressions of RRK and Weisskopf evaporation rates of this paper are the most commonly used. The RRK expression of evaporation rates used here, in particular, is not one of the modified versions introduced by some authors to avoid its intrinsic violation of the detailed balance principle [3,4]. The purpose of the present study is not to obtain accurate macroscopic extension of RRK and Weisskopf theories, but to give a rationale for their macroscopic extension and to stress the failure of the nonmicroreversible RRK model, which is unable to reproduce some properties of macroscopic systems such as equilibrium vapor pressures. The evaporation rate given by RRK-based models is an Arrhenius law of the form

  T k D A B  exp
where A does not depend on the temperature, whereas the Weisskopf model gives an expression of the form

  T k D T A B  exp
. Intermediate forms are also found in the literature, for instance [START_REF] Borggreen | [END_REF], Some authors, after having obtained the 1/T dependence of the prefactor in a microreversible scheme, finally give up this 1/T dependence and unjustifiably use a pure Arrhenius form with a constant prefactor [7],

  T k D T A B  exp
A practical conclusion of the paper is that, when a simple analytical expression is required to describe evaporation, the second form

  T k D T A B  exp
is preferable, especially when dealing with multiscale phenomena that runs from microscopic to macroscopic scales.

Microscopic models of unimolecular evaporation

The calculation of equilibrium vapor pressure requires describing the two basic phenomena involved, the evaporation of atoms from the condensed phase and the reverse process, the attachment of atoms onto condensed phase systems. Equilibrium between the condensed phase and the vapor is reached when attachment and evaporation probabilities are equal: the fulfillment of this condition is used here to calculate the vapor pressure as a function of temperature.

Two simple analytical expressions for the macroscopic equilibrium vapor pressure of a monoatomic compound are derived below, starting from the RRK model and the Weisskopf model, respectively, to calculate evaporation rates and using the hard sphere approximation to evaluate attachment rate. Simple versions of RRK and Weisskopf models are used, which do not include anharmonicity nor rotation, where the condensed phase is described as a collection of harmonic oscillators at the same frequency and the vapor is considered as a perfect gas. This simplicity allows calculating analytical expressions of Weisskopf and RRK evaporation rates, and of classical thermodynamics properties. The main difference between RRK and Weisskopf models lies in the fact that Weisskopf approach accounts for the translational density of states of the evaporated atom or molecule, whereas RRK theory which is an "activated complex theory", does not. As a consequence, RRK-based theories are unable to account for the entropy of the macroscopic vapor.

We will first recall the basics of RRK and Weisskopf models, then we will derive analytical expressions of equilibrium vapor pressure as a function of temperature from both models and compare these expressions to experimental equilibrium vapor pressures of sodium.

The RRK evaporation rate RRK model has been developed by Rice and Ramsperger [8] and, in parallel, by Kassel [9]. The basics of this well known theory are widely available in the literature [START_REF] Wendell Forst ; M | Unimolecular Reactions, a concise introduction[END_REF]. Let's just recall the main ideas: the dissociation of the so-called "activated complex" occurs as soon as the necessary energy is accumulated in one of the vibrational modes that lead to dissociation, called "critical oscillators". The dissociation probability is calculated from the ratio of the total number of quantum states of all activated complexes relative to the total number of states accessible to the system.

A system of n particles is represented as a collection of s=3n-6 quantum harmonic oscillators of the same frequency . The energy of each oscillator is h. The system contains a total energy E, whose distribution among the oscillators fluctuates randomly at a frequency close to . Evaporation occurs as soon as at least the dissociation energy D is accumulated on one particular oscillator, in other words when at least m>(D/h) quanta are focused on this critical oscillator. Let us call W n>m the total number of configurations that fulfill this condition, and W n the total number of configurations allowed for the system. The probability that a particular oscillator contains at least m quanta is P n>m =W n>m /W n [START_REF] Wendell Forst ; M | Unimolecular Reactions, a concise introduction[END_REF]. To obtain the probability that one atom evaporates from the whole cluster, this quantity is generally multiplied by the number of surface atoms n ext [START_REF] Zeifman | [END_REF]12,13]. If one considers that the internal energy fluctuates at the frequency , the evaporation probability per unit time is given by [START_REF] Ditchburn | For monoatomic vapors[END_REF] For large enough particles, the number of surface atoms of a spherical cluster containing n atoms is n ext (36) 1/3 n 2/3 [START_REF] Zeifman | [END_REF]12]; in order to calibrate in the following the parameters D and  from the experimental bulk equilibrium vapor pressure, it is useful here to state the evaporation rate as a function of the hard sphere cross section  n . The radius R of a cluster containing n atoms scales as r 0 n 1/3 , and the hard sphere cross section  n is equal to r 0 2 n 2/3 , thus n ext =(36) 1/3  n /(r 0 2 ).

Expression (1) is valid in the microcanonical ensemble, at a given total energy E. However, in the canonical ensemble, as far as the activation energy D is small compared with the mean total energy of the system E, the canonical rate constant [14]. The mean thermal internal energy is given here by

  T n k RRK evap , is well approximated by the microcanonical rate constant   E n k RRK evap ,
T sk E B  . Finally, for large n, using   x n n e n x     1 lim
, the RRK evaporation rate constant is given by [START_REF] Wendell Forst ; M | Unimolecular Reactions, a concise introduction[END_REF][START_REF] Zeifman | [END_REF]12]

              T k D r T n k B n RRK evap exp 36 , 2 0 3 1     (2)
The Weisskopf evaporation rate

    1 1 ,    s ext RRK evap E D n E n k



An alternative description of thermal evaporation, initially developed to describe neutron evaporation from nuclei, was developed by V. Weisskopf [15]. There is an important conceptual difference between both theories: While the RRK model is based on properties of the density of states of the "activated complex" and does not account for the properties of the dissociated product (when the evaporated atom or molecule is infinitely separated from the cluster), the Weisskopf model explicitly considers the density of state of the evaporated subunit. The Weisskopf theory satisfies the detailed balance principle: The evaporation rate

      n n atom n n n stick n n evap n E D E W E W             ) ( , 1 1 (3)
where  is the translationnal energy of the evaporated atom,

D

n is the dissociation energy (subscript n refer here and in the following to the number of atoms in the cluster), E n is the total internal energy of the cluster, and  n-1 ,  atom and  n are the density of states of the product cluster, the evaporated atom and the parent cluster, respectively. An analytic expression of the evaporation rate , where atom v is the velocity of the ejected atom and µ the reduced mass of the atom-cluster system; the attachment cross section is evaluated in the hard sphere approximation; rotational effects are neglected. Under these assumptions, the integration of relation (3) over all allowed energies  gives following expression for the evaporation rate per unit time of a cluster of size n containing an internal energy E [4]:

      1 2 3 1 1 8 ,       s s n n evap n E D E s E n k   (4)
where  n-1 is the attachment cross section of an atom onto the cluster of size n-1 and D n is the dissociation energy of a cluster containing n atoms.

             T k D T k m T n k B B n Weisskopf evap exp 1 8 , 3 1    (5)
When applied to very small systems, expressions (1) and ( 4) are of course to be corrected to take into account the size variation of D [2,16].

The bulk equilibrium vapor pressure

The equilibrium vapor pressure is calculated assuming that the number of evaporated atoms per unit time is the same as the number of atoms attached to the cluster during the same time. The attachment probability is deduced from the attachment cross section  n , calculated in the hard sphere approximation (thus independent of collision energy). The vapor of density  is described as a perfect gas. The attachment rate is given by:

  atom n stick n n k v   (6)
where v atom is the mean velocity of atoms. The bulk equilibrium vapor P sat is equal to  sat k B T in the perfect gas approximation. It is calculated here by setting

  T k k evap n stick n      . stick n k  
is given by relations [START_REF] Borggreen | [END_REF] and

  T k evap n  
is given by relations ( 2) and ( 5), in the RRK and Weisskopf approximations, respectively. Finally, the equilibrium vapor pressure

               T T k D mk r P B B RRK sat ln 2 1 exp 8 36 2 0 3 1     (7)                T T k D k m P B B Weisskopf sat ln 2 1 exp 2 3 2 3   (8)
An expression similar to relation ( 8) is implicitly given by Bertsch and coworkers [17]. The most important difference between both expressions, beyond the difference in the prefactors (that are independent of temperature), is the temperature dependence of the last term in the exponential factor, equal to

  T ln 2 1 
in the RRK expression and to

  T ln 2 1 
in the Weisskopf expression. This difference in the temperature dependence of both models directly arises from the introduction of the translational density of states of the evaporated atoms in the Weisskopf model, thus it is closely related to the Weisskopf theory obeying detailed balance principle while RRK theory does not. Moreover, it is shown in appendix that the Weisskopf expression of Another difference between Weisskopf and RRK-derived expressions of P sat lies in their different sensitivity to the energy dependence of the sticking cross section, which is not taken into account here. The equilibrium vapor pressure is less sensitive to the energy dependence of the sticking cross section when the Weisskopf model (or any microreversible model) is used, since the evaporation rate is "naturally" proportional to the sticking cross section, so that when writing the equilibrium condition (sticking rate = evaporation rate), the cross section (and thus its possible energy dependence) vanishes within the approximations used in this paper. This might contribute to the almost surprisingly good agreement of Weisskopf expression of vapor pressures with experimental data. This argument does not hold in the RRK approach, where the sticking cross section is "artificially" introduced in the expression of evaporation rate (see relation ( 2)).

A widely used semi-empirical expression for vapor pressures is the Kirchhoff's expression

C T B T A P     ln ln
, where A,B and C are constant parameters that are tabulated. It is worth noting that B is usually a positive constant (to very few exceptions such as helium) [18], which is consistent with expression (8) where B=+1/2, while it is in contradiction with expression (7) where B=-1/2.

Comparison with experiment of calculated vapor pressure of sodium

We take here the following Kirchhoff-type approximation of experimental vapor pressure of sodium Figure 1 shows the ratio of the values given by expression (9) to the values calculated using relations (7) and (8), in the RRK and Weisskopf approximations, respectively. The comparison spans the whole temperature range from sodium melting temperature (370 K) up to its critical temperature (2500K). The value of r 0 used in relation ( 7) is 2Å [16]. The two free parameters  and D are determined by a best fit approximation of expression (9) [16]; the exact value of  is not crucial in modelizing clusters evaporation rates).

Figure 1 shows very clearly that the Weisskopf expression of vapor pressure given by relation (8) is in much better agreement with experiment that RRK's one given by relation (7). The discrepancy between the RRK curve and the experimental curve is much larger than the uncertainty on the approximation given by relation (9) of experimental data [START_REF] Fink | Thermodynamic and transport properties of sodium liquid and vapor[END_REF].

Conclusion

The calculation of bulk vapor pressure from RRK and Weisskopf models, respectively, has been exemplified here in the case of sodium, for which small clusters evaporation properties are well documented. It is shown that the bulk equilibrium vapor pressures of sodium calculated using Weisskopf model are in better agreement with experimental data than the ones obtained using RRK model. It is demonstrated that the violation of detailed balance principle by RRK theory makes it inconsistent with classical thermodynamics and is certainly responsible of the poor agreement with the experimental data. The free parameters of microscopic evaporation models, namely the vibrational frequency of the cluster  and its dissociation energy D , can be deduced from experimental equilibrium vapor pressures. The values found for sodium using Weisskopf model,



, are very realistic. The method used here to deduce them from experimental equilibrium vapor pressures is a convenient way to calibrate them. The variation with temperature of the latent heat of evaporation given by classical thermodynamics is the same as the one deduced from Weisskopf theory, whereas it is not compatible with the one deduced from RRK theory (see appendix). Contrary to the RRK analytical expression of the bulk vapor pressure, the Weisskopf expression of the vapor pressure is compatible with the Kirchhoff expression for most compounds. Finally, it is shown that, when a simple expression is needed, an expression of the form

  T k B T A B  exp
seems to be preferable to an expression of the form

  T k B A B  exp
(where A and B do not depend on temperature in both cases) to evaluate the evaporation probability of a substance, especially when multiscale reliability from microscopic scale to macroscopic scale is required.

Appendix

The temperature dependence of the latent heat of evaporation L(T) can be calculated from Weisskopf and RRK evaporation rate and using the Clausius-Clapeyron relation. It is shown below that L Weisskopf (T) deduced from the Weisskopf expression of evaporation rate is in agreement with classical thermodynamics (at the same approximation level as in the Weisskopf model concerning the description of the cluster and the vapor), whereas L RRK (T) is not. Considering the vapor as a perfect gas and neglecting the variation of volume of the condensed phase, the Clausius-Clapeyron gives the following relation between the latent heat of evaporation L and the temperature dependence of equilibrium vapor pressure

1 

 1 must be linked by the following relation:

  from relation(3) under the following approximations[4]:  n (and similarly  n- 1 ) is calculated assuming that, as in the RRK calculation presented above, clusters are represented by s=3n-6 harmonic oscillators at the same frequency ; the density of states of the evaporated atom atom  is assumed to be the translational density of states of a free particle of kinetic energy  in a volume V,

  , equation (4) becomes:

  Weisskopf models, respectively, are given by the following relations:

Fig. 1 :

 1 Fig. 1: (Colour on-line) ratio of calculated to experimental sodium vapor pressures. The (red online) solid line is obtained using Weisskopf model of evaporation and the (blue online) dashed line is obtained using RRK model of evaporation. The theoretical parameters  and D are determined from the best fit of experimental vapor pressures given by relation (9):

  this relation, the expression of L Weisskopf and L RRK can be deduced from relations(7) and(8), respectively: thermodynamics, in the perfect gas approximation, gives the following temperature dependence of the latent heat: calculated for a cluster-perfect gas system from the following relations: Pierre Labastie for stimulating discussions and thorough advices.