Alexandre Santinho

Aymeric Chorlay

Lionel Foret

Abdou Rachid Thiam
email: thiam@ens.fr

Fat inclusions strongly alter membrane mechanics

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Introduction

Organisms from protists to humans store excess energy in carbon-rich molecules that are often neutral lipids (NLs) [START_REF] Murphy | Mechanisms of lipid-body formation[END_REF][START_REF] Van Wijk | Plastoglobuli: plastid microcompartments with integrated functions in metabolism, plastid developmental transitions, and environmental adaptation[END_REF]. NLs are apolar fat molecules, different from phospholipids, which are the building blocks of cellular membranes. The most abundant NLs are triacylglycerols, which are structurally the closest NL to phospholipids, and sterol esters, which are the most relative to cholesterol, another membrane constituent [START_REF] Walther | Lipid droplets and cellular lipid metabolism[END_REF]. However, other NLs are found depending on cell type and metabolic condition, such as retinyl ester, phytyl esters, squalene, acylceramides, etc. [START_REF] Van Wijk | Plastoglobuli: plastid microcompartments with integrated functions in metabolism, plastid developmental transitions, and environmental adaptation[END_REF][START_REF] Ajat | Hepatic stellate cells retain the capacity to synthesize retinyl esters and to store neutral lipids in small lipid droplets in the absence of LRAT[END_REF][START_REF] Senkal | Ceramide is metabolized to acylceramide and stored in lipid droplets[END_REF][START_REF] Spanova | Effect of lipid particle biogenesis on the subcellular distribution of squalene in the yeast Saccharomyces cerevisiae[END_REF][START_REF] Thiam | Lipid Droplet Nucleation[END_REF]. Most of these NLs are synthesized in the bilayer of the endoplasmic reticulum (ER) [START_REF] Olzmann | Dynamics and functions of lipid droplets[END_REF][START_REF] Pol | Review: biogenesis of the multifunctional lipid droplet: lipids, proteins, and sites[END_REF]. Because of their hydrophobicity, NLs accumulate within the bilayer's hydrophobic region as dissolved, mobile hydrophobic inclusions [START_REF] Santinho | Membrane Curvature Catalyzes Lipid Droplet Assembly[END_REF]. When their concentration reaches a critical value, NLs phase separate within the bilayer and nucleate an oil lens [START_REF] Thiam | The physics of lipid droplet nucleation, growth and budding[END_REF][START_REF] Khandelia | Triglyceride blisters in lipid bilayers: implications for lipid droplet biogenesis and the mobile lipid signal in cancer cell membranes[END_REF], which grows and buds off to become an independent organelle called a lipid droplet (LD) [START_REF] Olzmann | Dynamics and functions of lipid droplets[END_REF]. In the context of nucleation phenomena, the formation of an LD consists of packaging a large fraction of the NLs into a condensed phase, in equilibrium with a dilute NL phase in the bilayer [START_REF] Thiam | Lipid Droplet Nucleation[END_REF][START_REF] Thiam | The physics of lipid droplet nucleation, growth and budding[END_REF].

Avoiding the accumulation of NLs in the ER membrane ensures its proper functioning. NL accumulation could increase the bilayer's thickness, which would likely cause membrane stresses and affect the stability of transmembrane proteins. Indeed, such an accumulation may alter membrane structure, fluidity, lipid distribution, and stability, which can cause various forms of membrane stress. Most of these disturbances can explain the onset of diseases or cell death when NL molecules accumulate in the ER bilayer and are not properly removed and packaged into LDs.

The assembly of triacylglycerol LDs is facilitated by membrane curvature, composition, and specific proteins [START_REF] Santinho | Membrane Curvature Catalyzes Lipid Droplet Assembly[END_REF][START_REF] Adeyo | The yeast lipin orthologue Pah1p is important for biogenesis of lipid droplets[END_REF][START_REF] Ben M'barek | ER Membrane Phospholipids and Surface Tension Control Cellular Lipid Droplet Formation[END_REF][START_REF] Choudhary | Seipin and Nem1 establish discrete ER subdomains to initiate yeast lipid droplet biogenesis[END_REF][START_REF] Joshi | Lipid droplet and peroxisome biogenesis occur at the same ER subdomains[END_REF][START_REF] Kassan | Acyl-CoA synthetase 3 promotes lipid droplet biogenesis in ER microdomains[END_REF][START_REF] Wang | Seipin and the membrane-shaping protein Pex30 cooperate in organelle budding from the endoplasmic reticulum[END_REF] that decrease the critical nucleation concentration [START_REF] Santinho | Membrane Curvature Catalyzes Lipid Droplet Assembly[END_REF][START_REF] Chung | LDAF1 and Seipin Form a Lipid Droplet Assembly Complex[END_REF][START_REF] Zoni | Seipin accumulates and traps diacylglycerols and triglycerides in its ring-like structure[END_REF][START_REF] Prasanna | Seipin traps triacylglycerols to facilitate their nanoscale clustering in the ER membrane[END_REF].

Accordingly, when the ER's curved structure is lost due to the dysfunction in the ER shaping proteins, triacylglycerols accumulate in the bilayer, unable to effectively condense into LDs [START_REF] Santinho | Membrane Curvature Catalyzes Lipid Droplet Assembly[END_REF][START_REF] Wang | Seipin and the membrane-shaping protein Pex30 cooperate in organelle budding from the endoplasmic reticulum[END_REF]. These conditions are associated with several diseases, such as spastic paraplegia [START_REF] Renvoisé | Reep1 null mice reveal a converging role for hereditary spastic paraplegia proteins in lipid droplet regulation[END_REF].

Other factors such as the ER membrane phospholipid composition and surface tension can also regulate the concentration of the bilayer-dissolved NLs [START_REF] Ben M'barek | ER Membrane Phospholipids and Surface Tension Control Cellular Lipid Droplet Formation[END_REF] and their propensity to condense into LDs [START_REF] Zoni | Lipid droplet biogenesis is driven by liquid-liquid phase separation[END_REF]. Likewise, the NL chemistry determines its tendency to remain in the bilayer or condense into an LD [START_REF] Santinho | Membrane Curvature Catalyzes Lipid Droplet Assembly[END_REF][START_REF] Ben M'barek | ER Membrane Phospholipids and Surface Tension Control Cellular Lipid Droplet Formation[END_REF]. Squalene, an intermediate molecule in the synthesis of sterols, is an example of an NL that abnormally accumulates in membranes [START_REF] Spanova | Effect of lipid particle biogenesis on the subcellular distribution of squalene in the yeast Saccharomyces cerevisiae[END_REF][START_REF] Ben M'barek | ER Membrane Phospholipids and Surface Tension Control Cellular Lipid Droplet Formation[END_REF][START_REF] Spanova | Influence of squalene on lipid particle/droplet and membrane organization in the yeast Saccharomyces cerevisiae[END_REF]. Increasing the proportion of squalene relative to triacylglycerols leads to the formation of larger LDs [START_REF] Ben M'barek | ER Membrane Phospholipids and Surface Tension Control Cellular Lipid Droplet Formation[END_REF], probably because squalene increases the nucleation concentration [START_REF] Spanova | Effect of lipid particle biogenesis on the subcellular distribution of squalene in the yeast Saccharomyces cerevisiae[END_REF][START_REF] Spanova | Influence of squalene on lipid particle/droplet and membrane organization in the yeast Saccharomyces cerevisiae[END_REF] and decreases the budding propensity of LDs [START_REF] Ben M'barek | ER Membrane Phospholipids and Surface Tension Control Cellular Lipid Droplet Formation[END_REF]. Interestingly, many anticancer and antifungal drugs target the squalene epoxidase [START_REF] Petranyi | Allylamine derivatives: new class of synthetic antifungal agents inhibiting fungal squalene epoxidase[END_REF][START_REF] Brown | The shape of human squalene epoxidase expands the arsenal against cancer[END_REF][START_REF] Cirmena | Squalene epoxidase as a promising metabolic target in cancer treatment[END_REF], leading to reduced cellular cholesterol and, inevitably, an accumulation of squalene. These two consequences may concomitantly contribute to cellular death.

In this paper, we ask the fundamental question of how NL molecules, referred to as fat or oil molecules hereafter, modulate the mechanical properties of a phospholipid bilayer. To answer this question, we used model membranes based on the formation of giant unilamellar vesicles (GUVs) containing neutral lipids, such as those from droplet-embedded vesicles [START_REF] Chorlay | An asymmetry in monolayer tension regulates lipid droplet budding direction[END_REF][START_REF] Chorlay | Membrane Asymmetry Imposes Directionality on Lipid Droplet Emergence from the ER[END_REF] or inverted emulsion vesicles [START_REF] Pautot | Production of unilamellar vesicles using an inverted emulsion[END_REF]. We found that oil molecules' presence substantially modifies the bilayer response to stretching and bending, depending on the phospholipid composition and oil type.

Results

Oil droplets expand the critical bilayer area strain while decreasing permeability.

We opted for osmotic swelling as a straightforward means to probe the expansion capacity of a giant unilamellar vesicle (GUV) bilayer [START_REF] Chabanon | Pulsatile lipid vesicles under osmotic stress[END_REF][START_REF] Oglęcka | Oscillatory phase separation in giant lipid vesicles induced by transmembrane osmotic differentials[END_REF][START_REF] Sandre | Dynamics of transient pores in stretched vesicles[END_REF][START_REF] Karatekin | Cascades of transient pores in giant vesicles: line tension and transport[END_REF] (Figure 1A, top). To investigate if NLs alter this expansion, we used the droplet-embedded vesicle system (DEV) [START_REF] Chorlay | Making Droplet-Embedded Vesicles to Model Cellular Lipid Droplets[END_REF], consisting of a giant unilamellar vesicle (GUV) containing an oil droplet made of triolein, the major triacylglycerol NL (Figure 1A, bottom). In the DEV system, the oil droplet equilibrates within seconds with a constant fraction of triolein molecules mobile in the bilayer (Figure S1A-E) [START_REF] Santinho | Membrane Curvature Catalyzes Lipid Droplet Assembly[END_REF].

We prepared GUVs and triolein oil-based DEVs (Figure 1B, bottom), both made of DOPC phospholipid, and we subjected them to the same osmotic shock. Both systems showed swelling and bursting cycles during which their bilayer underwent stretching and relaxation phases [START_REF] Chabanon | Pulsatile lipid vesicles under osmotic stress[END_REF] (Figure 1A-B). To describe and understand the systems' response, we focused on the bilayer surface area strain 𝓔𝓔 for both GUVs and DEVs (Figure 1C). For DEVs, the bilayer area is calculated at each time point by withdrawing the space occupied by the droplet from the area of the sphere fitting the bilayer (Figure 1C, Figure S1F-I). Furthermore, the DEVs' swelling time took a few minutes (Figure 1C,D), much longer than the time it takes for triolein oil molecules to partition between the droplet and the bilayer [START_REF] Santinho | Membrane Curvature Catalyzes Lipid Droplet Assembly[END_REF]. Therefore, the swelling process of DEVs occurred at equilibrium.

Triolein-DEVs had a slower strain rate than GUVs (initial slope, Figure 1D), which underwent many cycles within the same time-lapse. The strain rate is determined by the water flux, i.e., the permeability of the bilayer: the higher the permeability, the higher the strain rate.

Accordingly, we found a permeability higher for GUVs, 24.8±8.3µm.s -1 , than triolein-DEVs, 9.0±2.6µm.s -1 (Figure 1E left, Figure S1J-M, supplemental text) [START_REF] Olbrich | Water permeability and mechanical strength of polyunsaturated lipid bilayers[END_REF]. Squalene-DEVs were less permeable than triolein-DEVs (Figure 1E left, Figure S1J,L,M, supplementary text), consistent with permeability studies with droplet interface bilayers [START_REF] Faugeras | Membrane determinants for the passive translocation of analytes through droplet interface bilayers[END_REF].

The critical strain 𝓔𝓔max, reached when the bilayer bursts, was higher for DEVs than GUVs (Figure 1D, Figure 1E right, Figure S1J). A burst occurs when a critical pore size appears in the membrane. GUVs stretched at a slower strain rate have a higher probability of reaching the critical pore size [START_REF] Chabanon | Pulsatile lipid vesicles under osmotic stress[END_REF] and, therefore, a lower 𝓔𝓔max. This correlation is not verified with DEVs, which had slower strain rates than GUVs while higher 𝓔𝓔max (Figure 1D, Figure 1E

right).

Therefore, other factors absent in GUVs favor the increase in 𝓔𝓔max for DEVs. We hypothesized that oil molecules recruited to the bilayer are responsible for this higher critical bilayer strain for DEVs.

To test our hypothesis, we decided to vary the phospholipid composition, which controls the bilayer's oil concentration [START_REF] Ben M'barek | ER Membrane Phospholipids and Surface Tension Control Cellular Lipid Droplet Formation[END_REF]. We worked with different phospholipid mixtures, including DOPE (PE), DOPA (PA), DPPC, or PufaPC (18:0, 20:4). For each condition, we determined the critical bilayer strain, 𝓔𝓔max, and compared it between GUVs and DEVs (Figure 1F, Figure S1N-R). Irrespective of composition, triolein-DEV bilayers withstood larger expansions than GUVs (Figure 1F, Figure S1R). Interestingly, the critical bilayer strain of triolein-DEVs was systematically higher in the presence of negatively curved lipids such as PA and PE, between 3 and 5 times higher than for GUVs (Figure 1F, Figure S1R). We found that squalene enabled a higher membrane expansion than triolein in PC conditions and, more strikingly, in PufaPC (Figure 1F, Figure S1R). This particular increase in PufaPC may pertain to π-π interactions between the numerous unsaturations in the squalene's carbon skeleton and those in the PufaPC acyl chain. However, compositions containing PA and PE showed no difference between the two oils (Figure 1F, Figure S1R), suggesting that these phospholipids' contribution masked the difference between the two oils.

Stretching increases the bilayer oil concentration depending on the phospholipid composition.

Based on the above results, we hypothesize that the oil droplet buffers oil molecules to the bilayer, which has more material to withstand more significant expansions. Following this hypothesis, PA and PE would lead to more oil molecules in the bilayer.

To test our model, we used a fluorescently-labeled "triolein", triolein-NBD, mixed with triolein at 0.05% (w/w) to report for triolein localization (Figure 2A). The resulting DEV was then monitored during the osmotic swelling (Figure 2B, left). We found that the bilayer's triolein signal increased concomitantly with the bilayer stretching (Figure 2B, right). Then, the bilayer concentration sharply decreased following a rapid burst of the DEV (Figure 2B, right); the oil molecules almost spontaneously re-equilibrated between the droplet and the bilayer after this burst (Figure 2B, Figure S2A). These results show that increasing the bilayer surface tension by osmotic swelling raises the amount of oil in the bilayer, agreeing with previous in silico results [START_REF] Ben M'barek | ER Membrane Phospholipids and Surface Tension Control Cellular Lipid Droplet Formation[END_REF] and confirmed here by using micropipettes to vary surface tension (Figure S2B,C).

To test if squalene behaves like triolein, we used Bodipy, which labels the bilayer's hydrophobicity, as we could not find available fluorescent squalene reporters. We observed a stronger Bodipy signal in the bilayer during osmotic swelling, most likely due to an increase in squalene in the bilayer (Figure S2D).

Our hypothesis suggests that the number of oil molecules incorporated into the bilayer during swelling determines the extent of membrane expansion. Since the critical bilayer strain depended on the phospholipid type (Figure 1F, Figure S1R), we measured the triolein increase in the bilayer for each phospholipid condition. We found an increase in the triolein signal during swelling for all lipid compositions but to different extents (Figure 2D, Figure S2E). We defined the bilayer triolein enrichment as the ratio between the triolein level in the bilayer at the maximum expansion (high tension) and the initially unstretched bilayer (low tension) (Figure 2C). We found that DEVs containing PA or PE, which had the most remarkable expansion (Figure 1F), displayed the highest triolein enrichment (Figure 2D). In contrast, triolein-DEVs made with PufaPC-PC, which did not significantly increase the critical bilayer strain (Figure 1F), exhibited a low triolein enrichment. These data are consistent with our model that, during stretching, triolein molecules can behave as membrane building blocks, like phospholipids, and allow for more bilayer expansion.

If our model is correct, then the enrichment level of triolein in the bilayer should dictate the critical bilayer strain. To test this prediction, we considered for each phospholipid composition the ratio of the critical bilayer strain of DEVs over GUVs (Figure 1F). This normalization cancels the contribution of phospholipids to the membrane expansion and highlights the contribution of triolein molecules. Consistent with our prediction, we found that the amount of triolein injected into the bilayer during stretching increases linearly with the critical bilayer strain ratio (Figure 2E).

In summary, an oil droplet connected to a bilayer equilibrates with residual oil molecules in the bilayer. The concentration of this residual oil increases as the bilayer is stretched. Phospholipids with negative spontaneous curvature favor a higher delivery of oil molecules from the droplet to the stretched bilayer and, thereupon, a higher critical bilayer strain.

The sole presence of oil molecules in a bilayer increases the critical area strain.

To fully demonstrate that the oil molecules favored the increase in the critical bilayer strain, we decided to repeat the previous experiments with oil molecules in the bilayer without any droplet reservoir. To do so, we used the water-in-oil inverted emulsion method (30) by using a triolein oil phase containing triolein-NBD (Figure 3A, left). With this approach, indeed, we found that the resulting GUVs, called triolein-GUV (To-GUV), carried only triolein molecules in their bilayer (Figure 3A, right). The bilayer triolein concentration is defined as the triolein-NBD bilayer signal divided by Cy5-PE, reporting phospholipids (Figure 3B). Based on this normalization, triolein-GUVs and -DEVs composed of PC, and not subjected to an osmotic swelling, had a similar bilayer triolein signal (Figure 3C).

We first compared GUVs and triolein-GUVs, both made of PC, subjected to osmotic swelling.

We found that the critical bilayer strain, 𝓔𝓔max, for triolein-GUVs was significantly higher than for GUVs (Figures 3D), consolidating our model that the sole presence of NLs is sufficient to increase the membrane expansion limit.

We next compared triolein-GUVs to triolein-DEVs. Triolein-DEVs had a 1.45 fold higher membrane expansion limit than triolein-GUVs in PC (Figure 3D). The presence of PA exacerbated this difference, by 3.2 fold higher (Figure 3E, S3A-B). These observations also agree with our model. Indeed, the bilayer of triolein-GUVs had a fixed number of oil molecules and, therefore, less material to withstand larger expansions; in contrast, the bilayer of triolein-DEV can be continuously supplied with oil molecules by the droplet. PA highlighted further this difference because it induced the most substantial triolein release into the bilayer of DEVs (Figure 2D-E). Finally, squalene had a similar impact than triolein in squalene-GUVs vs. squalene-DEVs (Figure 3F).

Overall, these data show that the oil molecules' presence increases the bilayer's ability to withstand more extensive stretching than a pure phospholipid membrane. If an NL droplet is physically connected to the bilayer, it will buffer NLs to the bilayer, further increasing its expansion. A dramatic consequence of such an increase in the bilayer stretching is that the area per phospholipid can be increased by several folds by NLs, especially in the presence of negatively curved phospholipids (Figure 1F, Figure . 3D-F).

Oil molecules modify the mechanical properties of a bilayer.

We wanted to know whether the mechanical properties of the bilayer are influenced by the presence of oil, as for solvents [START_REF] Needham | Tensions and free energies of formation of" solventless" lipid bilayers. Measurement of high contact angles[END_REF]. The critical bilayer surface area strain is determined by the appearance of a critical pore size, which depends on membrane stretching and bending factors [START_REF] Chabanon | Pulsatile lipid vesicles under osmotic stress[END_REF]. We probed how NLs altered these mechanical factors independently.

First, we studied if NLs altered the lysis tension of DEVs, i.e., the surface tension at which a pore opens in the bilayer. We used micropipette suction to increase the surface tension of PC GUVs and PC DEVs made of triolein or squalene. The micropipette aspiration (MPA)-applied bilayer tension was increased continuously by pulling on the bilayer at a rate of approximately 2-3 mbar/sec (Figure S4A). The lysis tension was taken as the MPA-applied bilayer tension was recorded right before the bilayer's burst (Figure S4A). In PC, the lysis tension of GUVs was 8.1 ± 1.9 mN/m, in the range of previously reported values [START_REF] Olbrich | Water permeability and mechanical strength of polyunsaturated lipid bilayers[END_REF][START_REF] Evans | Dynamic tension spectroscopy and strength of biomembranes[END_REF]. Triolein increased this lysis tension to 9.7 ± 1.9 mN/m, and squalene DEVs led to a more significant increase, to 10.8 ± 2.2 mN/m (Figure 4A). A similar trend was found for a PC/PA membrane composition (Figure S4B). This increase in the lysis tension can partly explain the higher critical bilayer area strain proffered by oil molecules (Figure 1F, Figure S1R).

Secondly, we determined the impact of the oil molecules on the area expansion modulus. For surface tensions over 0.5 mN/m, sufficient to smoothen thermal membrane fluctuations, the apparent area expansion modulus is given by Hooke's law [START_REF] Kwok | Thermoelasticity of large lecithin bilayer vesicles[END_REF][START_REF] Rawicz | Effect of chain length and unsaturation on elasticity of lipid bilayers[END_REF] (Figure 4B).

Experimentally, we increased the bilayer area with a micropipette while recording its MPAapplied surface tension (Figure 4B). The slope of the resulting MPA-applied bilayer tensionarea curve (Figure S4C-D) yields the apparent area expansion modulus of the bilayer (Yapp) [START_REF] Rawicz | Effect of chain length and unsaturation on elasticity of lipid bilayers[END_REF]. With this approach, we could only work with GUVs containing oils (Figure 3A) since, in DEVs, the droplet has a contribution in elasticity difficult to model. GUVs made of PC had an apparent expansion area modulus of 194 ± 25 mN/m (40), while those bearing triolein and squalene had lower values, 89 ± 31 mN/m and 111 ± 27 mN/m respectively (Figure 4C). This marked decrease in the apparent area expansion modulus is consistent with the overall increase in the critical bilayer strain observed when the NL oils are present in the bilayer (Figure 3D-E). However, no significant difference was found between squalene and triolein, suggesting that the apparent area expansion modulus does not solely account for the disparity in the bilayer expansion limit between these two oils (Figure 1F).

Finally, we determined the bending rigidity (K) by nanotube pulling experiments (Figure 4D and S4E-I). We found that the presence of triolein in a PC GUV bilayer decreased the bending rigidity (12 ± 4 kBT) roughly by a factor of two compared to the value for a bare GUV (21 ± 4 kBT) (Figure 4E); a similar trend was found when measuring the rigidity by fluctuation analysis [START_REF] Pakkanen | Mechanics and dynamics of triglyceride-phospholipid model membranes: Implications for cellular properties and function[END_REF]. Surprisingly, squalene slightly increased the bending rigidity (27 ± 4 kBT), illustrating that oil molecules do not necessarily change membrane properties in the same way. We used the bending rigidity values to compute the absolute area expansion moduli (Figure 4F) [START_REF] Olbrich | Water permeability and mechanical strength of polyunsaturated lipid bilayers[END_REF]. PC GUVs had an expansion area modulus of 214 ± 25 mN/m (40), while GUVs containing triolein or squalene had a lower value of 101 ± 35 mN/m and 120 ± 28 mN/m (Figure 4F).

In conclusion, the above data illustrate how and which membrane mechanical properties are altered by NL oil molecules. NLs can have similar or opposite effects on membrane mechanical properties, depending on the NLs' chemistry.

Discussion

Precise control of the biophysical properties of membranes is crucial for cell survival.

Phospholipids and membrane proteins often control these properties. Here, we consider the impact of neutral lipids, which can behave as hydrophobic membrane inclusions. We found that neutral lipids proffer unusual mechanical properties to the bilayer membrane. Such an effect depends on the nature of the hydrophobic inclusion but also the phospholipid composition.

We worked with triolein, the main component of cellular LDs, and squalene, which can significantly accumulate in cells upon drug treatments targeting the squalene epoxidase [START_REF] Petranyi | Allylamine derivatives: new class of synthetic antifungal agents inhibiting fungal squalene epoxidase[END_REF][START_REF] Brown | The shape of human squalene epoxidase expands the arsenal against cancer[END_REF][START_REF] Cirmena | Squalene epoxidase as a promising metabolic target in cancer treatment[END_REF]. For the phospholipid conditions that we tested, we found that these neutral lipids increased the lysis tension of a bilayer and significantly decreased its area expansion modulus. While triolein decreased the bilayer bending stiffness, squalene increased it. Overall, these mechanical alterations are consistent with the greater membrane expansion proffered by the neutral lipids.

At the molecular level, the increased critical membrane strain likely results from the insertion of neutral lipids between phospholipids when the membrane is stretched, i.e., when tension increases. Thus, the neutral lipids may adopt different molecular conformations when tension varies [START_REF] Hamilton | Interactions of triglycerides with phospholipids: incorporation into the bilayer structure and formation of emulsions[END_REF][START_REF] Pakkanen | Phase coexistence in a triolein-phosphatidylcholine system. Implications for lysosomal membrane properties[END_REF][START_REF] Bacle | Interdigitation between triglycerides and lipids modulates surface properties of lipid droplets[END_REF]. At low tension, triolein molecules could remain perpendicular to the phospholipids but parallel to them at high tension, exposing their glycerol backbone to water [START_REF] Pakkanen | Phase coexistence in a triolein-phosphatidylcholine system. Implications for lysosomal membrane properties[END_REF]. This molecular adaptability would allow phospholipids to be spaced further apart than in a pure bilayer.

Squalene is a long terpene with numerous unsaturation, chemically different from triolein. Despite contrasting chemistry, the two neutral lipids yielded similar elasticity. As for triolein, squalene may also adopt phospholipid-parallel or antiparallel orientations in the bilayer [START_REF] Tarun | Interaction of oil and lipids in freestanding lipid bilayer membranes studied with label-free high-throughput wide-field second-harmonic microscopy[END_REF], depending on the surface tension, in a way that decreases the area expansion modulus.

Concerning the membrane lysis tension, both neutral lipids increased it, especially squalene. This difference may pertain to the interaction spectrum of NLs with phospholipids. In particular, squalene can make multiple interactions π-π (47) with phospholipids, making it act like a molecular glue, "cross-linking" the bilayer leaflets and delaying the opening of a pore.

Interestingly, squalene increased the bilayer bending rigidity, whereas triolein decreased it, relative to GUVs.

The bending rigidity (κ) depends on the bilayer thickness (d), the area expansion modulus (Y), and the coupling constant (ß) of the monolayer leaflets composing the bilayer: κ ∝ ß.d 2 .Y [START_REF] Bermudez | Effect of bilayer thickness on membrane bending rigidity[END_REF].

Triolein and squalene decreased by two, the area expansion modulus, Y, (Figure 4F), which tends to reduce the bending rigidity, but not for squalene (Figure 4E). Squalene particularly increases membrane thickness, more than triacylglycerols [START_REF] Faugeras | Membrane determinants for the passive translocation of analytes through droplet interface bilayers[END_REF][START_REF] Needham | Tensions and free energies of formation of" solventless" lipid bilayers. Measurement of high contact angles[END_REF][START_REF] Gross | Determining membrane capacitance by dynamic control of droplet interface bilayer area[END_REF][START_REF] Waldbillig | Planar bilayer membranes from pure lipids[END_REF]. Also, via π-π interactions developed with phospholipids [START_REF] Hildebrandt | Interactions between phospholipids and organic phases: insights into lipoproteins and nanoemulsions[END_REF], squalene likely increases the monolayer coupling ß. Consequently, the contribution of ß.d 2 for squalene may compensate for the decrease in Y, sufficiently to overall increase the bending rigidity κ.

Based on the above analysis and the rigidity measurement (Figure 4E), curving squalene-rich membranes is more difficult. Squalene accumulation is promoted by several antifungal and anticancer drugs [START_REF] Petranyi | Allylamine derivatives: new class of synthetic antifungal agents inhibiting fungal squalene epoxidase[END_REF][START_REF] Brown | The shape of human squalene epoxidase expands the arsenal against cancer[END_REF][START_REF] Cirmena | Squalene epoxidase as a promising metabolic target in cancer treatment[END_REF] and, therefore, the subsequent alteration in membrane mechanics could contribute to the death of fungal or cancer cells. Inducing the synthesis and membraneaccumulation of specific neutral lipids could be a strategy to prompt cancer cells to die. Finally, our data show that a droplet connected to a bilayer can supply neutral lipids to the membrane, especially when the bilayer experiences stretching. The amount of neutral lipids delivered to the bilayer depends on the bilayer phospholipid composition. Thus, in a cellular context, LDs should be disconnected from the ER bilayer to avoid residual neutral lipids in the membrane. If not, the neutral lipid accumulation would hinder the ER 's multiple functions that rely on the precise regulation of its mechanical properties, involving membrane tubulation, stretching, budding, or fusion.

In conclusion, our study provides important insights into how neutral lipids can change membranes' biophysical properties. This study is relevant to better understand the ER membrane mechanics during fat storage, particularly in pathological conditions where neutral lipids fail to efficiently phase separate into LDs. Our approaches can be adapted to test the behavior of more exotic lipids. decreases apparent area expansion modulus. see also (Figure S4 C-E). D) Experimental determination of the bending modulus (K) by pulling a nanotube from the bilayer. The bending modulus is directly linked to the nanotube radius and the membrane tension. For further details see (Figure S4E-I). E) Bending modulus of GUVs (n=7), triolein-GUVs (n=4) and squalene-GUVs (n=3) made with PC. Mean +/-SD. The presence of triolein in the bilayer results in a significant bending modulus decrease.

Squalene increases the bending modulus. E) Area expansion modulus of PC GUVs (n=5), triolein-GUVs (n=7) and squalene-GUVs (n=5). Mean +/-SD. The presence of neutral lipids in the bilayer (triolein or squalene) decreases area expansion modulus.

MATERIALS AND METHODS

All experiments were performed in the following HKM buffer: 50 mM Hepes, 120 mM Kacetate, and 1 mM MgCl2 (in Milli-Q water) at pH 7.4 and 280±15 mOsm.

Preparation of GUVs.

GUVs were prepared by electro-formation. Phospholipids and mixtures thereof in chloroform at 0.5µM were dried on an indium tin oxide (ITO) coated glass plate. The lipid film was desiccated for 1 h. The chamber was sealed with another ITO coated glass plate. The lipids were then rehydrated with a sucrose solution (280+-15 mOsm). Electro-formation is performed using 100 Hz AC voltage at 1.0 to 1.4 Vpp and maintained for at least 2h. This low voltage was used to avoid hydrolysis of water and dissolution of the titanium ions on the glass plate. GUVs were either stored in the chamber at 4°C overnight or directly collected with a Pasteur pipette.

GUVs composition for used Phospholipids mixtures:

PC GUVs were 100 % PC, PE-PC GUVs were 50% PE and 50% PC, PA-PC GUVs were 30% PA and 70% PC, DPPC-PC GUVs were 40% DPPC and 60% PC, PufaPC-PC GUVs were 30% PufaPC and 70% PC.

Preparation of DEVs

To prepare the oil droplets, 5 µL of the oil (triolein or squalene) was added to 70 µL of HKM buffer. The mixture was sonicated. The diameter of the resulting droplets is on the order of a few hundred nanometers. To make DEV, GUVs were then incubated with the LDs for 5 min.

To make DEV with triolein-NBD, the oil was composed of 99,95% of triolein and 0,05% (w/w) of To-NBD. The GUV-LD mixture was then placed on a glass coverslip (pretreated with 10 % (w/w) BSA and washed three times with buffer) and observed with a confocal microscope.

Preparation of GUVs by the inverted emulsion technique

Two phospholipids mixtures were diluted in chloroform at 0.5 µM and dried. A 150µL oil volume was poured onto one mixture and 200µL of HKM-buffer onto the other one. These solutions were sonicated for 5 minutes to facilitates the solubilization of phospholipids. A 100µL volume of the oil-phospholipid mixture was deposited on the HKM-phospholipids solution. Then, the remaining 50µL of the oil-phospholipid mixture was mixed with 4µL of an HKM-sucrose 1:1 (w/w) solution to generate a water-in-oil inverted micro-emulsion which was then added to the HKM-contacting-oil solution. The resulting mixture was centrifuged for three minutes. The oil-containing GUVs were next collected in the aqueous volume.

Confocal microscope images

All micrographs of DEV were made on a Carl ZEISS LSM 800 with an X63 oil immersion objective or X10 air objective, and observed samples were held by Glass coverslips (Menzel Glaser (24x36mm #0)).

Hypotonic shock

DEVs were produced following the techniques described above. The resulting solution was placed on a coverslip (pretreated with 10 % (w/w) BSA and washed three times with buffer) and observed with a confocal microscope (x10 objective). Spherical DEVs with no thermal fluctuation of the bilayer have been selected so that the surface of the bilayer can be easily determined. Next, 50µL of MQ water was added to 120µL of DEV solution to applied an osmotic shock of ∆C=80mOsm. The osmotic gradients applied were always the same (∆C=80mOsm for Figure 1,2,3 and Figure S1,S2,S3) except for (Figure 1D,E and Figure S1J-M due to permeability protocol precautions) where the ∆C was 145mOsm and for (Figure 2B and Figure S2A to report rapid triolein partitioning) where ∆C was 210mOsm. The DEVs were then imaged overtime to follow the evolution during swelling. The same protocol was used to apply hypotonic shock to GUVs or Oil-containing-GUVs (Figure 1F,2D,3D,E,F).

Determination of the critical bilayer strain 𝓔𝓔max

The maximum expansion of the bilayer is given by the following equation:

𝜖𝜖 𝑚𝑚𝑚𝑚𝑚𝑚 = 𝐴𝐴 𝑚𝑚𝑚𝑚𝑚𝑚 -𝐴𝐴 0 𝐴𝐴 0
where Amax and A0 are respectively the maximal areas of the bilayer before burst (stretched state) and the area before starting the swelling experiments (unstretched state). This area was determined by fitting a circle onto the DEV (GUV) bilayer and recording its radius. The radius of the embedded droplets was also recorded for DEV swelling experiments to determine the corrected surface area of the DEVs bilayer (see the section below).

Bilayer surface area of the DEVs

To consider only the surface of the DEV's bilayer, the surface occupied by the embedded droplet had to be removed. To do this, we calculated the surface of the sphere formed by the bilayer and subtracted the surface of the spherical cap corresponding to the droplet (Figure S1C). This gave the following expression of the surface area of the DEV bilayer (A)

𝐴𝐴 𝐷𝐷𝐷𝐷𝐷𝐷 = 2𝜋𝜋𝑅𝑅 𝑔𝑔 2 . (1 + � 1 -( 𝑅𝑅 𝑑𝑑 𝑅𝑅 𝑔𝑔 ) 2 )
where Rg and Rd are respectively the DEV radius and droplet radius.

Permeability assay

To determine permeability, we made osmotic shock experiments thanks to an applied osmotic gradient between the outside and inside of the vesicles of ∆C=145mOsm. To do so, we added 20µL of DEVs (or GUVs) sucrose solution (280±15 mOsm) into 100µL HKM buffer. Then, we added 100µL of Milli-Q water to induce an osmotic shock. During the experiment, we maintained the DEVs (or the GUVs) thanks micropipettes, to have a clear determination of both vesicle radius and droplet radius (See Figure 1C, and supplemental material 1). This enables to determine the permeation area and the volume of the vesicle as a function of time during the swelling experiment.

GUVs, triolein-DEV,s and squalene-DEVs initial permeability computation

See supplemental material 1 for the computation of the initial permeability.

Determination of the triolein concentration in the bilayer and following of this concentration as a function of time

To determine the concentration of triolein molecules in the bilayer, we used triolein-DEV (triolein GUVs). The triolein mixture was made of 99,95% triolein and 0,05% (w/w) of triolein-NBD. For all our experimental conditions involving these measurements (

FRAP experiment, triolein-NBD bleaching in the Bilayer

FRAP experiments were performed by bleaching a part of the DEV bilayer (Figure S1B-C).

To-NBD signal was bleached, then, the fluorescence signal recovery was monitored. The FRAP curve is normalized by the fluorescence before bleaching DEVs and just after the bleach (in non-bleached regions). GraphPad Prism was used to fit the FRAP recovery curve with a nonlinear regression and the exponential "one-phase association model".

Bodipy fluorescence in squalene DEVs during hypotonic shock

To follow bodipy fluorescence signal during hypotonic shock upon squalene DEVs (Figure S2F), 0,5µL of bodipy was added in 100µL of HKM buffer containing Squalene DEVs. After 10 minutes, 20µL of water was added to the coverslip and the evolution of bodipy signal in the bilayer of the Squalene DEV was monitored.

Micromanipulation & surface tension measurements by micropipette aspiration applied bilayer tension (MPA-applied bilayer tension).

Micropipettes were made from capillaries drawn out with a Sutter Instruments pipette puller.

They were used to manipulate GUVs, DEVs, or GUVs containing oil. The pipettes were incubated for 30 min in a 5% BSA solution before use, to prevent droplets and membranes from adhering to the glass.

Additionally, surface tensions of bilayers were measured using the same pipettes. Using Laplace's law, and measuring the pipette inner radius, the GUV radius, and suction pressure, the surface tension of the bilayer can be determined by:

𝛾𝛾 = ∆P 𝑠𝑠𝑠𝑠𝑠𝑠 2 � 1 𝑅𝑅 𝑝𝑝 - 1 𝑅𝑅 𝑔𝑔 �
where ΔPsuc, Rp, and Rg are the suction pressure, the pipette inner radius, and the vesicle radius, respectively

The suction was carried out using a syringe. For bending rigidity measurements (see below, Bending rigidity measurements), that needed a better measurement precision, the suction was applied thanks to a water reservoir whose height was control with a micrometer gauge (precision 0.005 mm). The resulting suction pressure was determined thanks to the measurement of the water level variation.

Lysis tension measurements

To determine an estimation of the bilayer lysis tension of GUVs, DEVs, or GUVs containing oil, we used the micropipette aspiration technique. Thanks to a slight aspiration, a bilayer tongue was sucked in the micropipette. The aspiration was then increased at an approximate rate of 2-3 mbar/sec, causing a proportional increase in the bilayer surface tension (see above,

Micromanipulation & surface tension measurements).

At a certain tension, the bilayer ruptured because of a pore opening. The measured lysis tension was taken as the higher tension reached just before bilayer rupture.

Bending rigidity measurements

First, nanotubes were pulled from GUVs or oil-GUVs: Streptavidin-coated microspheres were added to the bulk on the glass coverslip. Then, the GUV was gently captured by one of the two micro-pipettes to control surface tension γ. One streptavidin-coated microsphere was caught by the other micro-pipette and it was slowly moved toward the GUV. Upon contact, the biotinylated phospholipid of the GUV interacted strongly with the streptavidin-coated microsphere surface. Finally, a nanotube was pulled by removing slowly the micropipette in the opposite direction to the GUV.

The bending rigidity was obtained with the following equation:

𝑅𝑅 𝑡𝑡 = � 𝜅𝜅 2𝛾𝛾
Where κ is the bending rigidity of the membrane which has to be determined with the experimental measurements of the bilayer tension γ (see Micromanipulation & surface tension measurements by micro-aspiration section) and nanotube radius Rt.

To obtain Rt, we vary the tube length at a constant displacement velocity of 2,5µm.s -1 at constant surface tension, which keeps the radius constant based on the previous equation.

Therefore, for a given tension, we have many couples of tube length Lt and tongue length Lp in the micropipette. The conservation of surface yields: 𝐿𝐿 𝑝𝑝 = 𝛿𝛿. 𝐿𝐿 𝑡𝑡 + 𝑏𝑏 Thus, from this conservation, the slope (𝛿𝛿) was extracted by measuring different couples of Lp and Lt. We also checked that we do not detect hysteresis on these curves (Figure S4H-I) to ensure that we reached equilibrium for the oil partitioning (for triolein-GUV and squalene-GUV) between the nanotube and the GUV.

On the other hand, conservation of the volume of the system links the radius of the tube to that of the GUV (Rg) and the pipette through the following equation ( 51):

𝑅𝑅 𝑡𝑡 = 𝛿𝛿. �1 - 𝑅𝑅 𝑝𝑝 𝑅𝑅 𝑔𝑔 � . 𝑅𝑅 𝑝𝑝
Finally, different values of Rt, associated with a different value of γ were measured, gave us a mean value for κ with the first equation of this section.

Apparent area expansion modulus measurements

GUVs or Oil-containing-GUVs were gently captured with a micropipette. Then, aspiration in the micropipette was slowly increased to obtain multiple values of membrane tension (approximately 20 sec between each point). For each value of MPA-applied bilayer tension the variation of the length of the bilayer tongue in the micropipette and the radius of the GUV was measured allowing to determine an approximate variation at the first order of the surface area of the bilayer of the GUV ΔA (36) :

ΔA = 2𝜋𝜋𝑅𝑅 𝑝𝑝 ( 1 -𝑅𝑅 𝑝𝑝 /𝑅𝑅 𝑔𝑔 )Δ𝐿𝐿 𝑝𝑝
where Δ𝐿𝐿 𝑝𝑝 , Rp, and Rg are respectively the variation of the bilayer tongue length in the micropipette, the pipette inner radius, and the GUV radius.

The MPA-applied bilayer tension was then plotted against the relative surface area variation of the GUV's bilayer. As the GUV was already slightly tensed (initial tension above 0.1 mN/m) the membrane fluctuation is negligible and the MPA-applied bilayer tension (𝛾𝛾) can be linked to the surface area variation (Δ𝐴𝐴/𝐴𝐴 0 ) by the apparent area expansion modulus (Y):

𝑌𝑌 𝑚𝑚𝑝𝑝𝑝𝑝 = 𝛾𝛾 Δ𝐴𝐴/𝐴𝐴 0
Therefore, the slope of the MPA-applied bilayer tension -surface area variation curve gave the apparent area expansion modulus of the membrane (Yapp).

Area expansion modulus computing

expansion modulus measurements" section and previously determined bending rigidity 𝜅𝜅. To plot the MPA-applied bilayer tension 𝛾𝛾 against the relative surface area variation Δ𝐴𝐴/𝐴𝐴 0 as above, the MPA-applied bilayer tension is the plot as a function of :

Δ𝐴𝐴 𝐴𝐴 0 + ∆𝛼𝛼 With ∆𝛼𝛼 = -(𝐾𝐾 𝐵𝐵 𝑇𝑇 8𝜋𝜋𝜅𝜅)ln (𝛾𝛾 𝛾𝛾 0 ) ⁄ ⁄
𝐾𝐾 𝐵𝐵 is the Boltzmann constant, T the room temperature in Kelvin and 𝛾𝛾 0 the initial MPA-applied bilayer tension at the beginning of the experiment. Therefore, the slope of the MPA-applied bilayer tension -Δ𝐴𝐴 𝐴𝐴 0 + ∆𝛼𝛼 curve gave the area expansion modulus (Y) [START_REF] Olbrich | Water permeability and mechanical strength of polyunsaturated lipid bilayers[END_REF].

QUANTIFICATION AND STATISTICAL ANALYSIS

Statistical analysis

The statistical comparison was made using student a parametric test where we assume a Gaussian distribution. Then the test is an unpaired t-test. The statistical analysis were made by GraphPad Prism 7.0a and "***" indicates p < 0.0005, "**" indicates p < 0,005 and "*" (M8266-100G), BSA 98% (A7906-100G), and sucrose 99.5% (59378-500G) were from Sigma-Aldrich.

Fat Inclusions Strongly Alter Membrane Mechanics.

Alexandre Santinho 1, § , Aymeric Chorlay 1, § , Lionel Foret 1 , Abdou Rachid Thiam 1 * 1 Laboratoire de Physique de l'École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France § These authors contributed equally to this work

C. GUV and DEV permeation surface: hypothesis and computation as a function of 𝐑𝐑 𝐠𝐠,𝐜𝐜

The permeation surface of the GUV ( 𝑆𝑆 𝑡𝑡 𝐺𝐺𝐺𝐺𝑉𝑉 ) corresponds to the surface of the sphere formed by its bilayer and can be expressed as:

𝑆𝑆 𝑡𝑡 𝐺𝐺𝐺𝐺𝑉𝑉 = 4𝜋𝜋𝑅𝑅 𝑔𝑔,𝑡𝑡 2
In the case of the DEV, the water cannot pass through the oil droplet. Thus to assess the permeation surface of a DEV ( 𝑆𝑆 𝑡𝑡 𝐷𝐷𝐷𝐷𝑉𝑉 ) we have to subtract the surface area occupied by the droplet 𝑆𝑆 𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐 from the surface area of the sphere forming its bilayer 𝑆𝑆 𝑡𝑡 𝐺𝐺𝐺𝐺𝑉𝑉 (see Figure 1) : Compared to an equivalent droplet-free vesicle, the permeation area of the DEV must therefore be corrected by the following factor:

𝑆𝑆 𝑡𝑡 𝐷𝐷𝐷𝐷𝑉𝑉 = 𝑆𝑆 𝑡𝑡 𝐺𝐺𝐺𝐺𝑉𝑉 -𝑆𝑆 𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐
𝛽𝛽 𝑡𝑡 = 𝑆𝑆 𝑡𝑡 𝐷𝐷𝐷𝐷𝑉𝑉 𝑆𝑆 𝑡𝑡 𝐺𝐺𝐺𝐺𝑉𝑉 = 1 2 �1 + �1 -( 𝑅𝑅 𝑑𝑑,𝑡𝑡 𝑅𝑅 𝑔𝑔,𝑡𝑡 ) 2 �
Correction factor 𝛽𝛽 𝑡𝑡 :

As the radius of the vesicle (𝑅𝑅 𝑔𝑔,𝑡𝑡 ) and the radius of the droplet (𝑅𝑅 𝑑𝑑,𝑡𝑡 ) evolved during the osmotic swelling (Figure 2), we calculated 𝛽𝛽 𝑡𝑡 over time and found that it only varied from 0,922 to 0,899. These variations are represented in Figure 3. In order to better quantify this small decrease of 𝛽𝛽 𝑡𝑡 , we calculated the ratio between the initial 𝛽𝛽 𝑡𝑡 and its value after 60 second of each osmotic swelling experiment (𝛽𝛽 60 /𝛽𝛽 0 , Figure 4, right).

We find that 𝛽𝛽 𝑡𝑡 has very small variations: 2,15 ± 0.67 % and 1,04 ± 0.69 % respectively for

To-DEV and Sq-DEV (Figure 4, right). Based on these observations, we considered 𝛽𝛽 𝑡𝑡 = 𝛽𝛽 0 .

We plotted all the value of 𝛽𝛽 0 in (Figure 4, left). For the volume of a GUV, Vt can be expressed as:

𝑉𝑉 𝑡𝑡 𝐺𝐺𝐺𝐺𝑉𝑉 = 4 3 𝜋𝜋𝑅𝑅 𝑔𝑔,𝑡𝑡 3
The inside DEV volume 𝑉𝑉 𝑡𝑡 𝐷𝐷𝐷𝐷𝑉𝑉 has to be corrected because of the presence to the droplet:

𝑉𝑉 𝑡𝑡 𝐷𝐷𝐷𝐷𝑉𝑉 = 𝑉𝑉 𝑡𝑡 𝐺𝐺𝐺𝐺𝑉𝑉 -𝑉𝑉 𝑡𝑡 𝑑𝑑𝑑𝑑𝑑𝑑𝑐𝑐,𝑖𝑖𝑖𝑖
where 𝑉𝑉 𝑡𝑡 𝑑𝑑𝑑𝑑𝑑𝑑𝑐𝑐,𝑖𝑖𝑖𝑖 is the lacking of aqueous volume in the vesicle occupied by the embedded droplet (see Figure S1A). 𝑉𝑉 𝑡𝑡 𝐷𝐷𝐷𝐷𝑉𝑉 can be expressed as: Where 𝛼𝛼 𝑡𝑡 is the correction factor of the DEV inner volume compared to the one of GUVs. As we did previously for 𝛽𝛽 𝑡𝑡 , 𝛼𝛼 𝑡𝑡 is plotted as a function of time (Figure 3). The variation of 𝛼𝛼 𝑡𝑡 is also very small, from 0,930 to 0,937. We also examined for each experiment the variations of In conclusion, the volume of permeation of a DEV and a GUV write as:

𝑉𝑉 𝑡𝑡 𝐷𝐷𝐷𝐷𝑉𝑉 = 4 3 𝜋𝜋𝑅𝑅 𝑔𝑔,𝑡𝑡 3 - 𝜋𝜋 3 (ℎ 𝑑𝑑𝑜𝑜𝑡𝑡,𝑡𝑡 2
𝛼𝛼
𝑉𝑉 𝑡𝑡 = 𝛼𝛼 0 . Here, 𝑃𝑃 𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑖𝑖𝑐𝑐𝑖𝑖 , "initial permeability" replaces the permeability because we considered only initial slope at the beginning of experiment. For a given experiment, < 𝑅𝑅 𝑔𝑔,𝑡𝑡 > is the mean of Rg,t in the first 60 seconds.

Figure 1 :

 1 Figure 1: Oil droplets connected to a bilayer expand the critical membrane area strain. A) Top: Schematic representation of the behavior of a GUV under hypotonic shock. GUVs undergo swelling and burst cycles. Bottom: Schematic representation of a triolein DEV (Droplet Embedded Vesicle) system compared to a GUV which also undergo swelling and burst cycles. B) Time-lapse of triolein DEV and GUV during a hypotonic shock experiment. Black dashed circles report maximum vesicle radius. Scale bar is 10µm. C) Top: Green arc underlying the bilayer area for an uninflated and inflated DEV. Bottom: Formula used to compute the bilayer strain, 𝓔𝓔, defined as the relative expansion of the bilayer area. D) Corresponding bilayer strain 𝓔𝓔 variations of triolein DEV and GUV over time of Figure 1B. Swelling and burst cycles are clearly visible for GUVs (blue points; 6 cycles) and DEVs (green points; 1 cycle). E) Left: Plot of the initial water permeability of PC bilayer of GUVs (n=8), triolein-DEVs (n=9) and squalene-DEVs (n=9). Mean +/-SD. Right: Plot of the critical bilayer strain (𝓔𝓔max) of GUVs (blue, n=28), triolein-DEVs (green, n=29) and squalene-DEVs (grey, n=21) for PC bilayer. Mean +/-SD. F) Plot of the critical bilayer strain (𝓔𝓔max) of GUVs (blue), triolein-DEVs (green) and squalene-DEVs (grey) for various phospholipid compositions of the bilayer. Mean +/-SD.

Figure 2 :

 2 Figure 2: Stretching increases the oil concentration in the bilayer depending on the phospholipid composition. A) Schematic representation of the triolein DEV formation protocol. To report triolein distribution, triolein was supplemented with 0,05% (w/w) triolein-NBD. B) Left, confocal images of PC triolein DEVs under hypotonic shock with triolein tagged by triolein-NBD. Line scans show a triolein-NBD signal colocalizing with the bilayer. Scale bar is 5µm.Right, corresponding quantification of triolein-NBD signal and bilayer area variation. During bilayer stretching, triolein-NBD signal increased, and quickly dropped after deflation. C) Representative confocal fluorescence images of a PC triolein-DEVs at the maximum bilayer expansion state (high tension) and at the "unstretched" state (low tension). Linescans allow for quantification of triolein-NBD fluorescence intensity in the bilayer in each state. 'Triolein enrichment' was computed by taking the fluorescence ratio between stretched state and unstretched state. Scale bar is 5 μm. D) Triolein enrichment measured for PC (n=8), PA-PC (n=5), PE-PC (n=4), DPPC-PC (n=3) and PufaPC-PC (n=8) triolein-DEVs. Mean +/-SD. E) Plot of the critical bilayer strain ratio between DEV and GUV against the triolein enrichment for previous phospholipid compositions. Statistical analysis revealed a significant slope different from zero for a linear fit (dotted lines). Mean +/-SD.

Figure 3 :

 3 Figure 3: The sole presence of oil molecules increases the critical bilayer strain. A) Left, illustration of the formation of triolein-GUVs (To-GUVs) using a water-in-oil inverted emulsion method. The triolein was tagged with 0,05% (w/w) of triolein-NBD. Right, fluorescence confocal imaging of the resulting triolein-GUVs. Significant triolein-NBD signal colocalize with the bilayer reported by Cy5-PE. Scale bar: 5µm. B) Top: three different model systems: a bilayer with no

Figure 4 :

 4 Figure 4: Oil molecules modify the mechanical properties of a bilayer. A) Bilayer lysis tension measured for GUVs (n=43), triolein-DEVs (n=24) and squalene-DEVs (n=41), made with PC. Mean +/-SD. B) Illustration of the experimental setup to determine the apparent area expansion modulus of a bilayer. Thanks to micropipettes, tension is measured while increasing the bilayer area. Apparent area expansion modulus (Y) is computed from the ratio between surface tension and area variation. C) Apparent area expansion modulus of PC GUVs (n=5), triolein-GUVs (n=7) and squalene-GUVs (n=5). Mean +/-SD. The presence of neutral lipids in the bilayer (triolein or squalene)

Figures 1 , 2

 12 and 3), we chose DEVs (triolein GUVs) which radius was between 5 and 15 microns with micrometric triolein droplets. DEVs snap were taken by keeping the same settings for each DEVs. Then, fluorescence profiles were drawn perpendicular to the bilayer of triolein-DEVs (triolein GUVs) and averaged over 15 pixels. The maximum value of the profiles was recorded and averaged over multiple DEVs. The same protocol was used for Cy5-PE fluorescence signals measurements (Figure3C). For monitoring of triolein concentration in the bilayer (FigureS1D-E), we captured images every 5 minutes to avoid bleaching of the triolein-NBD signal.

  indicates p < 0.05. When the bar graph is plotted, values shown in the text and figures are mean ± S.D. When only the data set of one experiment is shown, the values on the graph are absolute value ± uncertainty of measurements. EQUIPMENT AND REAGENTS Equipment All micrographs were made on a Carl ZEISS LSM 800. Glass coverslips: Menzel Glaser (24x36mm #0). Micro-pipettes made from capillaries (1.0ODx0.58IDx150Lmm 30-0017 GC100-15b Harvard Apparatus) with a micropipette puller (Sutter instrument model P-2000). Micromanipulation (Eppendorf TransferMan® 4r). Pressure measurement unit (DP103 provided by Validyne eng. corp, USA). Chemical products: Phospholipids: PC (1,2-dioleyl-sn-3glycero-3-phosphocholine), PE (1,2-dioleoyl-sn-glycero-3-phosphoethanolamine), PA (1,2-dioleoyl-sn-glycero-3-phosphate), DPPC (1,2-dipalmitoylsn-glycero-3-phosphocholine), PufaPC 18:0-20:4 (1-stearoyl-2-arachidonoyl-sn-glycero-3phosphocholine), biotynil-PE (1,2-dioleyl-sn-glycero-3-phosphoetanolamine-N-(biotinyl), Rh-PE (1,2-dioleyl-sn-glycero-3-phosphoethanolamine-N-lissamine rhodamine B sulfonyl) and Cy5-PE (1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-(Cyanine 5)) were from Avanti Polar Lipids. Bodipy 493/503 was from Thermo Fisher Scientific. Triolein and squalene were from Sigma Aldrich and To-NBD (1,3-Di(cis-9-octadecenoyl-2-((6-(7-notrobenz-2-oxa-1, 3diazol-4-yl) amino) hexanoyl) glycerol was from Setareh biotech. Streptavidin-coated microspheres were from Bangs Laboratories. Hepes (54457-250-F), K acetate (P1190), MgCl2

Figure S1: related to figure 1 A)

 1 Figure S1: related to figure 1 A) Confocal images of a PC GUV marked with Cy5-PE. The same settings as in Figure 2B were used for the green channel. No signal of triolein-NBD was visible here. Scale bar: 5µm. B) Confocal images of PC DEVs of triolein supplemented with 0,05% (w/w) triolein-NBD to report for triolein distribution. Image sequence of a FRAP (Fluorescence Recovery After Photobleaching) experiment on the bilayer. After bleaching the bilayer (see arrowhead), the triolein-NBD signal recovered. The scale bar is 5µm. C) Over time quantification of normalized triolein-NBD fluorescence signal after bleaching showing recovery. The non-linear fit matches the FRAP recovery curve. The fit was done with an exponential "one-phase association model". D) Confocal images of PC DEVs of triolein supplemented with triolein-NBD. The triolein DEVs were imaged for 25 minutes with no osmotic shock. The scale bar is 10µm. E) Over time variations of the triolein NBD-signal in the bilayer shows no variation of triolein concentration during 25 minutes. Absolute value +/-measurement uncertainty. F) Confocal bright field images captured at x10 objective of single PC DEV of triolein at a low tension state. Top view and side view showing the same droplet radius (Rd) used to compute the initial bilayer area of a DEV. The scale bar is 10µm. G) Confocal bright field images captured at x63 objective of single PC DEV of triolein at a high tension state with a flattened droplet. Top view and side view showing the same droplet radius (Rd) used to compute the final bilayer area of a DEV. The scale bar is 5µm. H) Top: Illustration of a DEV and needed parameters to compute the DEV bilayer area subtracting the area occupied by the embedded droplet. Bottom: Equation used to compute the bilayer area of a DEV. I) Top: Schematic representation of the bilayer area variations during osmotic swelling of a DEV (top view and side view). Green arcs underly the bilayer area. Bottom: Equation used to compute the bilayer strain 𝓔𝓔. J) Representative images of a PC squalene-DEV observed at x63 objective during a hypotonic shock experiment. The first image corresponds to the unswollen state and the second to one of the critical bilayer strain before burst. Dashed circles report maximum radius. Scale bar: 10µm. K) Related to Figure 1B. The plot of the relative vesicle radius variations of triolein DEV and GUV over time. Swelling and burst cycles are visible for GUVs (blue points; 6 cycles) and DEVs (green points; 1 cycle). Absolute value +/-measurement uncertainty. L) Plot of the vesicle relative radius variation at the beginning of a swelling experiment of a GUV (blue points), a triolein DEV (green points), and a squalene DEV (grey points). The slope of the linear fit (dotted lines on the graph) was used to determine the initial radius increase rate. Absolute value +/-measurement uncertainty. M) Plot of the initial radius increase rate of PC bilayer of GUVs (n=8), triolein-DEVs (n=9), and squalene DEVs (n=9). Mean +/-SD. N) Time-lapse of triolein DEV and GUV during a

Figure S2: related to figure 2 Santinho

 2 Figure S2: related to figure 2

Figure S3: related to figure 3 A)

 3 Figure S3: related to figure 3 A) Confocal images of PA-PC triolein-DEV and triolein-GUV with triolein-NBD. Scale bar: 5µm. B) Plot of the triolein in bilayer for PC-PA GUVs (n=4), triolein-GUVs (n=9), and triolein-DEVs (n=22). Mean +/-SD.

Figure 1 :

 1 Figure 1: triolein-DEVs of DOPC at x63 objective showing parameters to compute the corrected surface and volume of a DEV.

Figure 2 :

 2 Figure 2: Plot of Rg and Rd as a function of time

Figure 3 :

 3 Figure 3: Plot of the correction coefficient 𝛼𝛼 and 𝛽𝛽

Figure 4 :

 4 Figure 4: Left: Mean value of 𝛽𝛽 for DEVs permeability assay; Right: Variations of 𝛽𝛽 during permeability experiment.

1 . 3 =

 13 �3𝑅𝑅 𝑔𝑔,𝑡𝑡 -ℎ 𝑑𝑑𝑜𝑜𝑡𝑡,𝑡𝑡 � + (ℎ 𝑖𝑖𝑖𝑖,𝑡𝑡 2 (3𝑅𝑅 𝑑𝑑𝑑𝑑𝑑𝑑𝑐𝑐,𝑡𝑡 𝑖𝑖𝑖𝑖 -ℎ 𝑖𝑖𝑖𝑖,𝑡𝑡 ) where ℎ 𝑑𝑑𝑜𝑜𝑡𝑡,𝑡𝑡 , ℎ 𝑖𝑖𝑖𝑖,𝑡𝑡 and 𝑅𝑅 𝑑𝑑𝑑𝑑𝑑𝑑𝑐𝑐,𝑡𝑡 𝑖𝑖𝑖𝑖 are all defined in Figure 𝑡𝑡 -ℎ 𝑑𝑑𝑜𝑜𝑡𝑡,𝑡𝑡 � + (ℎ 𝑖𝑖𝑖𝑖,𝑡𝑡 2 (3𝑟𝑟 𝑑𝑑𝑑𝑑𝑑𝑑𝑐𝑐,𝑡𝑡 𝑖𝑖𝑖𝑖 -ℎ 𝑖𝑖𝑖𝑖 , 𝑑𝑑) 4𝑅𝑅 𝑔𝑔,𝑡𝑡 𝛼𝛼 𝑡𝑡 For the experiment in figure 2 the variation of 𝛼𝛼 𝑡𝑡 are plotted as a function of time (Figure 3). Santinho & al. 2021 Biophysical Journal Correction factor 𝛼𝛼 𝑡𝑡 :

  𝑡𝑡 during the first 60 seconds 𝛼𝛼 60 /𝛼𝛼 0 (Figure5, right) which were very small: 1,11 ± 0.35 % and 0,50 ± 0.48 % respectively for To-DEV and Sq-DEV (Figure5, right). Based on these observations, we considered 𝛼𝛼 𝑡𝑡 = 𝛼𝛼 0 . We plotted all the value of 𝛼𝛼 0 in (Figure5 left).

Figure 5 :

 5 Figure 5: Left: Mean value of 𝛼𝛼 for DEVs permeability assay; Right: Variations of 𝛼𝛼 during permeability experiment.

  With this expression, we determined the initial permeability to water Pinitial for PC GUVs, PC triolein-DEVs and PC squalene-DEVs. (Principal Figure1E, left)

Figure 1 ε

 1 Figure 1

Figure 4

 4 Figure 2

  Considering the expression Δ𝐶𝐶(𝑑𝑑) established in B. one obtains:Replacing 𝑉𝑉 𝑡𝑡 𝑎𝑎𝑎𝑎𝑑𝑑 𝑆𝑆 𝑡𝑡 by their expression obtained in C. and D leads to:We obtain a differential equation of the first order with a constant second member:

	𝑑𝑑(	𝑅𝑅 𝑔𝑔,𝑡𝑡 𝑅𝑅 𝑔𝑔,0 𝑑𝑑𝑑𝑑	)	+	3𝑃𝑃. 𝛽𝛽 0 . 𝛼𝛼 0 -1 . 𝑣𝑣 𝑚𝑚 . 𝐶𝐶 0 𝑖𝑖 𝑅𝑅 𝑔𝑔,0	𝑅𝑅 𝑔𝑔,𝑡𝑡 𝑅𝑅 𝑔𝑔,0	=	𝑃𝑃. 𝛽𝛽 0 . 𝛼𝛼 0 -1 . 𝑣𝑣 𝑚𝑚 . 𝐶𝐶 0 𝑖𝑖 𝑅𝑅 𝑔𝑔,0	. �4 -	𝑒𝑒 𝐶𝐶 0 𝐶𝐶 0 𝑖𝑖 �
	So:
		𝑑𝑑� 𝑅𝑅 𝑔𝑔,0 𝑅𝑅 𝑔𝑔,𝑡𝑡 𝑑𝑑𝑡𝑡	�	+ 𝑎𝑎	𝑅𝑅 𝑔𝑔,𝑡𝑡 𝑅𝑅 𝑔𝑔,0	=	𝑐𝑐 3 �4 -	𝐶𝐶 0 𝑒𝑒 𝐶𝐶 0 𝑖𝑖 � where 𝑎𝑎 =	3𝑃𝑃.𝛽𝛽 0 .𝛼𝛼 0 -1 .𝑣𝑣 𝑚𝑚 .𝐶𝐶 0 𝑖𝑖 𝑅𝑅 𝑔𝑔,0
	As we have linear curve for	𝑅𝑅 𝑔𝑔,𝑡𝑡 𝑅𝑅 𝑔𝑔,0	for the first 60 seconds (Figure S1L),	𝑑𝑑� 𝑅𝑅 𝑔𝑔,0 𝑅𝑅 𝑔𝑔,𝑡𝑡 𝑑𝑑𝑡𝑡	�	= 𝜔𝜔 is constant
	and equal to the slope of			𝑅𝑅 𝑔𝑔,𝑡𝑡 𝑅𝑅 𝑔𝑔,0		.
		𝜔𝜔 = (�4 -	𝐶𝐶 0 𝑒𝑒 𝐶𝐶 0 𝑖𝑖 � -3	< 𝑅𝑅 𝑔𝑔,𝑡𝑡 > 𝑅𝑅 𝑔𝑔,0	)	𝑃𝑃 𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑖𝑖𝑐𝑐𝑖𝑖 . 𝛽𝛽 0 . 𝛼𝛼 0 -1 . 𝑣𝑣 𝑚𝑚 . 𝐶𝐶 0 𝑖𝑖 𝑅𝑅 𝑔𝑔,0
							𝑑𝑑𝑉𝑉 𝑡𝑡 𝑑𝑑𝑑𝑑	= -𝑃𝑃. 𝑆𝑆 𝑡𝑡 . 𝑣𝑣 𝑚𝑚 . 𝐶𝐶 0 𝑖𝑖 . � 𝐶𝐶 0 𝑒𝑒 𝐶𝐶 0 𝑖𝑖 + 3	𝑅𝑅 𝑔𝑔,𝑡𝑡 𝑅𝑅 𝑔𝑔,0	-4�
	𝑑𝑑(𝛼𝛼 0 .	4 3 𝑑𝑑𝑑𝑑 𝜋𝜋𝑅𝑅 𝑔𝑔,𝑡𝑡	3 )	= -𝑃𝑃. 𝛽𝛽 0 . 4𝜋𝜋𝑅𝑅 𝑔𝑔,𝑡𝑡	2 . 𝑣𝑣 𝑚𝑚 . 𝐶𝐶 0 𝑖𝑖 . � 𝐶𝐶 0 𝑒𝑒 𝐶𝐶 0 𝑖𝑖 + 3

4 3 𝜋𝜋𝑅𝑅 𝑔𝑔,𝑡𝑡 3 for DEVs and 𝑉𝑉 𝑡𝑡 = 4 3 𝜋𝜋𝑅𝑅 𝑔𝑔,𝑡𝑡 3 for GUVs E. Fick law with 𝐑𝐑 𝐠𝐠,𝐜𝐜 as the sole variable:

Santinho & al. 2021 Biophysical Journal

ACKNOWLEDGMENTS

We are thankful to all the group members, Chloë Thimonier and Jean Goder, for their valuable comments and critical read of the manuscript. This work was supported the ANR-NANODROP, ANR-17-CE11-0003, ANR-MOBIL, ANR-18-CE11-0012-01 to ART.

Competing interests

The authors declare no competing interests.

Author contributions

ART designed research. AC and AS performed experiments and analyzed the data with LF. AC, AS, and ART wrote the manuscript. Santinho & al. 2021 Biophysical Journal

Supplementary materials: initial permeability determination for GUVs and DEVs

A. 𝐅𝐅𝐅𝐅𝐅𝐅𝐅𝐅 𝐥𝐥𝐥𝐥𝐥𝐥: 𝐅𝐅𝐜𝐜𝐜𝐜𝐜𝐜𝐅𝐅𝐜𝐜𝐅𝐅𝐜𝐜𝐜𝐜𝐜𝐜 𝐜𝐜𝐨𝐨 𝐮𝐮𝐜𝐜𝐅𝐅𝐥𝐥𝐅𝐅𝐮𝐮𝐥𝐥𝐜𝐜𝐅𝐅𝐜𝐜𝐜𝐜

Diffusion times of solutes through a phospholipid bilayer are negligible compared to the diffusion time of water. We therefore considered the diffusion of solutes through the bilayer of GUVs (or DEVs) as negligible compared to the diffusion of water. In this condition, Fick law leads to:

where 𝑉𝑉 𝑡𝑡 is the volume inside the vesicle delimitated by 𝑆𝑆 𝑡𝑡 the area of permeation (ie area of bilayer), P the bilayer permeability to water, 𝑣𝑣 𝑚𝑚 the molar volume of water (18g.mol -1 ) and Δ𝐶𝐶(𝑑𝑑) the difference of solutes concentration between the outside 𝐶𝐶 𝑒𝑒 (𝑑𝑑) and the inside 𝐶𝐶 𝑖𝑖 (𝑑𝑑) Where Rg,t and Rg,0 are respectively the vesicle radius at the time t and at the beginning of swelling (Figure 1). As we have small variations of radius (<10%). We made the approximation that:

𝑅𝑅 𝑔𝑔,𝑡𝑡 = 𝛿𝛿𝑅𝑅 𝑔𝑔,𝑡𝑡 + 𝑅𝑅 𝑔𝑔,0

where 𝛿𝛿𝑅𝑅 𝑔𝑔,𝑡𝑡 represent the small variations of vesicle radius.

It follows that:


Nicoletta Noceti
email: noceti@disi.unige.it

Matteo Santoro
email: santoro@disi.unige.it

Francesca Odone
email: odone@disi.unige.it

Unsupervised Learning of Behavioural Patterns for Video-Surveillance

come   L'archive ouverte pluridisciplinaire

Introduction and Background

Classication based on dynamic information is a challenging problem in computer vision and plays a central role in a number of present-day applications including visual surveillance, human-machine interface, and semantic retrieval from videos [START_REF] Hu | A survey on visual surveillance of object motion and behaviors[END_REF][START_REF] Zhang | Active and dynamic information fusion for facial expression understanding from image sequences[END_REF][START_REF] Efros | Recognizing action at a distance[END_REF][START_REF] Robertson | Behaviour understanding in video: a combined method[END_REF]. Although the urging need for eective systems in all these contexts is fuelling an intense research activity, several crucial issues are still either not solved or even not addressed. Specically in the eld of visual surveillance, with which we are mainly concerned in this paper, an important issue is the ecient analysis of behavioural patterns of people activities in the scene.

Given the huge variety of possible behaviours in moderately complex scenarios, a fully supervised classication approach is likely to fail to produce systems that are both robust i.e. able detect and recognise eectively many classes of events and exible, i.e. easy to be modied to recognise new (classes of ) events.

The peculiarity of video-surveillance is a great help in this direction. Usually a system observes the same scene for very long periods. Thus, such system could learn common behaviours in that scene from long-time observations, carried out in an initial or intermediate calibration stage. This aim calls for some form of unsupervised data analysis that does not depend on data labelling and does not require any a priori decision on the classes of interest. An analysis of available literature makes it clear that unsupervised methods for behaviour understanding or, more in general, to time series analysis are quite limited -see [START_REF] Jebara | Spectral clustering and embedding with hidden markov models[END_REF][START_REF] Liao | Clustering of time series data: A survey[END_REF][START_REF] Niebles | Unsupervised learning of human action categories using spatial-temporal words[END_REF][START_REF] Makris | Learning semantic scene models from observing activity in visual surveillance[END_REF].

Many successful methods have been proposed to discover classes of similar pattern in a dataset. Unfortunately most of them are not tailored for clustering sequential data, such as temporal series. In fact, while analysing temporal data, the temporal dimension usually induces specic, inherent structure to the sequence of feature vectors that is likely to be disregarded by traditional clustering methods. Furthermore, sequences of dierent lengths cannot be compared in a unique way. Previous works on clustering temporal series approached the problem from two distinct viewpoints (see [START_REF] Liao | Clustering of time series data: A survey[END_REF] and references therein). The rst approach relies on modifying suitable existing algorithms for clustering static data in such a way that time series data can be handled. An alternative is to convert temporal series data into the form of static data so that the existing algorithms for clustering static data can be directly used, trying to keep information on the structure of data. Our work is more related to this latter approach.

One of the main contributions of this paper is to set out an analogy between behaviour analysis for visual surveillance systems and the unsupervised classication of temporal/sequential series. Since our primary objective is to develop ecient strategies for analysing the behaviour of people moving in a scene, this analogy should allow for a protable deployment of successful approaches to temporal series analysis. The underlying idea is that recent advances in unsupervised learning technologies may play a major role in developing robust yet exible visual surveillance systems. In particular, we argue that spectral clustering-based algorithms achieve good performances in analysing the behaviours once suitable similarity measures have been chosen for comparing temporal series. Within this general framework we discuss an algorithmic architecture, based on unsupervised learning, for building a module of a video-surveillance system that learns automatically usual behaviours in a scene of interest from the observation of the scene itself. Understanding what is common or usual in a scene sets the basis for the design of anomaly detection systems.

The processing pipeline we refer to in this paper has been eectively described in [START_REF] Hu | A survey on visual surveillance of object motion and behaviors[END_REF]: rst, moving objects are detected (motion segmentation) and possibly classied in the scene. Second, single entities/objects previously detected, are tracked along the video sequence in order to build a model of the dynamic evolution of the trajectory as long as the identity of the object can be preserved.

Third, the trajectories are analysed in order to understand and describe the behaviours. Finally, information coming from multiple cameras, if available, is registered and combined. In this paper we focus on the third step, hypothesising that video inputs have been conveniently pre-processed insofar that a number of feature vector sequences are computed and then associated to objects moving in the scene. The choice of the specic feature map by means of which to represent a specic moving object at each time instant is not crucial to our analysis: we only require that the sequences are described by means of d×T matrices, where d is the number of components of the feature map and T is the number of elements in the sequence. In general, T may change from sequence to sequence.

In order to fully exploit spectral clustering algorithm, we focus on two basic issues: [START_REF] Hu | A survey on visual surveillance of object motion and behaviors[END_REF] how data should be represented, and (2) which is a proper similarity measure to compare them. In Section 2 dierent representation schemas are described and appropriate kernel functions are plugged into the general spectral clustering algorithm. An experimental analysis on synthetically generated data (Sec. 3) shows that the combination of a language-based representation and string kernels are the most appropriate for the analysed application domain. This combination is then applied to nd behaviour patterns in datasets of real image sequences used as a benchmark by the video-surveillance community. Preliminary results on such datasets speak in favour of the appropriateness of our approach, as discussed in Section 4.

The Approach

In this section we describe an algorithmic schema for the unsupervised analysis of behavioural patterns associated to image sequences. After reviewing some popular approaches to the representation of temporal series, we discuss the spectral clustering-based algorithm to discover behavioural patterns and describe suitable kernel functions to deal with data characterised by a variable length and an internal ordering due to temporal coherence. Finally, we motivate empirically the choices.

Representation of temporal series

We denote X = {x i } N i=1 the dataset of N temporal series, where each datum x i is a sequence of k i vectors in some Euclidean space R d , i.e. x i = (x i,1 , x i,2 , . . . , x i,ki ) T and x i,t ∈ R d . A natural way to interpret the index t is as the temporal index. The most popular approaches to describe information contained in X are the following:

1. Non-Parametric Representation of Temporal Series. If a non-parametric approach is adopted, then the representation of a set of temporal series is simply X itself. For instance, in the special case where all the series have the same length -

k i = k ∀i -it is sometimes convenient to think of a k × (d • N ) matrix X = (x 1 , x 2 , . . . , x N ).
This allows for a compact formulation of some structural properties of the data in terms of vector matrix calculations. When the lengths k i are not all the same, zero-padding or resampling [START_REF] Johnson | Learning the distribution of object trajectories for event recognition[END_REF] techniques may be applied or, more commonly, parametric representations may be considered. In the following we review a set of possible parametric approaches.

Parametric representation of temporal sequences by means of curve tting.

In a parametric setting, one aims at representing an event by means of a xed-length feature vector in order to overcome two crucial issues related to temporal series: [START_REF] Hu | A survey on visual surveillance of object motion and behaviors[END_REF] the fact that the events to be represented do not last the same amount of time, and (2) they do not have the same spatio-temporal evolution (sometimes even within events of the same class).

A popular and easy-to-implement parametric approach is based on curve tting, in which a regression function is built for each sequence (see [START_REF] Pittore | Learning to recognize visual dynamic events from examples[END_REF] and references therein). Here an important step is to choose a specic family of tting functions. Such choice depends on the distribution of the points in the scene, which is strongly connected to the complexity of observed behaviours.

Common alternatives are polynomials for moderately complex series [START_REF] Lin | Temporal data mining using multilevellocal polynominal models[END_REF] or a family of spline models for more general and complex scenario [START_REF] Pittore | Learning to recognize visual dynamic events from examples[END_REF].

As described below, in our experiments we used second-degree polynomials tted by means of a simple least squares-based method.

3. Parametric representation of temporal sequences as probability models. In a probabilistic setting, temporal sequences modelling may consist of building a probabilistic model of the present observation given all past observations: p(x i,t |x i,1 , y i,2 , . . . , y i,t-1 ). Because the history of observations can grow arbitrarily large it is necessary to limit the complexity of such a model. A common approach to overcome this problem is to limit the window of past observations. The simplest model, p(x i,t |x i,t-1 ), is known as a rst-order Markov model. One popular choice of probabilistic models that make this assumption, are the hidden Markov models (HMMs), of which it is always possible to write down explicitly the likelihood function [START_REF] Rabiner | A tutorial on hidden Markov models and selected applications in speech recognition[END_REF]:

p(x|θ) = q0,...,q k p(x 0 |q 0 )p(q 0 ) k t=1 p(x t |q t )p(q t |q t-1 ) (1)
where the q = {q 1 , . . . , q k } are the hidden states corresponding to each element of x and can take on one of the discrete values {1, . . . , M }. In order to specify an HMM, one has to specify the parameter vector θ = (π, α, µ, Σ), where π ∈ R M represents the initial state probability distributions π i = p(q 0 = i) for all i = 1, . . . , M . α ∈ R M ×M is the state transition probability distribution, i.e. α ij = p(q t = j|q t-1 = i). µ = {µ 1 , . . . , µ M } and Σ = {Σ 1 , . . . , Σ M } represents the mean and covariance of the Gaussian distributions for each state i, i.e. the emission density p(x t |q t = i) is a normal distribution N (x t |µ i , Σ i ).

Parametric Representation of Temporal Sequences as Strings. A sequence x

can be also considered a concatenation of symbols from a nite alphabet A.

As for the choice of the alphabet we consider a partition of the d-dimensional input space. Fully relying on an unsupervised approach to the problem we estimate an appropriate partition from the available dataset. To this purpose we cluster data in the d-dimensional input space, disregarding the temporal component -i.e. {{x i,t } t=1,...,ki } i=1,...,N .

In our experiments spectral clustering is used to cluster data. As for the kernel choice, o-the-shelf kernels, such as the Gaussian kernel, may be used.

Out-of-sample extension produces a full partition of the space. Once the alphabet is built, a temporal series x may be translated into a string s with an association of each element x j ∈ R d to the partition it belongs to. To obtain compressed descriptions that capture the peculiarities of each behaviour we consider only transitions between states, skipping the replicates of the same symbol, i.e., if a string contains replicates of the symbol u ∈ A it can be compressed as: αu λ β → αuβ, for every arbitrary substrings α and β.

2.2

Clustering of temporal series Spectral clustering refers to a class of techniques to partition points into disjoint clusters by analysing the eigenstructure of the Laplacian matrix of a similarity graph associated to the dataset. In [START_REF] Chung | Spectral graph theory[END_REF], the rst few eigenvectors of the graph's Laplacian are shown to carry information about the optimal cut to partition the graph. Dierent algorithms have been proposed in the last years that exploit this result (see [START_REF] Ng | On spectral clustering: Analysis and an algorithm[END_REF] and [START_REF] Shi | Normalized cuts and image segmentation[END_REF] for two popular examples). For the experiments, we implemented an ecient recursive algorithm for spectral clustering following [START_REF] Shi | Normalized cuts and image segmentation[END_REF]. With this approach we do not have to decide the number of the clusters beforehand, but we have only to dene the minimum size allowed for the small clusters. The level of granularity is specied through a threshold (CT ) on the value of the cut. The partitioning is stable across multiple runs of the algorithm because we do not need a nal k-means step as in [START_REF] Jebara | Spectral clustering and embedding with hidden markov models[END_REF] (where the authors slightly adapted the algorithm in [START_REF] Ng | On spectral clustering: Analysis and an algorithm[END_REF]).

It is well assessed that, in the spectral approach, clustering results strongly depend on the ability to capture the underlying notion of metric -or the similarity structure -over the input space. In turn, the similarity structure is related to the choice of a proper kernel function on the data. A function to be used as a kernel for spectral clustering should be symmetric and positive. Since in our experiments we compare dierent representation schemes we have to choose appropriate kernels. Both in the non-parametric and in the polynomial-based setting we refer to well established kernels such as the Gaussian kernels. The other two approaches require a more careful choice of the kernel able to deal with variable length data [START_REF] Taylor | Kernel Methods for Pattern Analysis[END_REF]. We considered the following:

1. Probability Product Kernel (PPK). An ecient measure of similarity among temporal series represented by means of HMMs, introduced in [START_REF] Jebara | Probability product kernels[END_REF], and used in [START_REF] Jebara | Spectral clustering and embedding with hidden markov models[END_REF] in tasks similar to ours.

Given two probability distributions representing two pairs of data sequences

over the space of all potential observable sequences X, the generalised inner product can be computed by integrating the product of the distributions:

K(p(x|θ), p(x|θ)) = X p β (x|θ)p β (x|θ)dx, (2)
where β is a free parameter which allow for the specication of some properties of the kernel. For instance, if β = 1/2, the PPK becomes the classic Bhattacharyya anity metric between two probability distributions, which is a Mercer kernel and can be conveniently used in our context. In addition, it is computable in closed form for a variety of distribution families. For the HMMs, we used the guidelines proposed in [START_REF] Jebara | Spectral clustering and embedding with hidden markov models[END_REF].

2. P-spectrum Kernel. If one interprets the temporal series as strings over a nite alphabet, then it is possible to borrow from studies on text and biological data manipulation [START_REF] Leslie | The spectrum kernel: a string kernel for svm protein classication[END_REF][START_REF] Lodhi | text classication using string kernels[END_REF]. The most natural way to compare two series is to count how many transition they have in common from one symbol of the alphabet to a dierent. This is tightly related to the concept of spectrum of order P (or P-specturm) of a sequence s to be the histogram of frequencies of all its (contiguous) substrings of length P . In the case of simple transitions between atomic symbols of the alphabet, we are considering basic substrings of length 2. Formally, the kernel may be dened as a feature map of strings followed by an appropriate dot product [START_REF] Taylor | Kernel Methods for Pattern Analysis[END_REF]. The map makes explicit all sub-strings of length P of string s:

φ P u (s) = |{(v 1 , v 2 ) : s = v 1 uv 2 }|, u ∈ A P . (3)
The associated kernel between two strings s and t is dened as:

K P (s, t) = φ P (s), φ P (t) = u∈A P φ P u (s)φ P u (t). (4)
String length independence is achieved with by a normalisation [START_REF] Taylor | Kernel Methods for Pattern Analysis[END_REF].

Experimental Assessment

In the previous section we introduced a general algorithm for unsupervised analysis of temporal series and proposed a number of variants for data representation. By leveraging on some preliminary experimental results, we can now discuss more thoroughly the properties of each specic choice and validate the eectiveness of the overall approach

For the experiments, we generated a number of temporal series of 2D points on a plane -see Figure 1. Such data are evocative of a simple yet realistic video surveillance scenario, as shown in Fig. 2), where positions of objects moving along the image plane are used to describe dynamic events. In this case the input space is so that d = 2, The sequences are divided into 5 groups with an increasing level of noise (0.1, 0.3, 0.5, 0.7, 0.9 respectively), so to model situations where a smooth object tracking is increasingly more dicult (e.g., because of scene clutter or bad illumination).

Each group of series comprises 3 distinct behavioural models (represented with dierent colours in Figure 1). In order to make some statistical analysis of the results, we resampled 20 dierent versions of each group of data. Since in principle some of the proposed kernels should be able to capture more complex properties of temporal series -such as to distinguish between opposite directions -we generated a second database (named DS6B) where each of the 3 behaviours is splitted in two subgroups by choosing randomly the direction of the points. Note that in our setting this is equivalent to switch the temporal index. The output of a simple people detection and tracking system from the database PETS06 (see later for details). The gure shows that our synthetic dataset can be conveniently used as a testbed for the proposed method.

-Gaussian Kernels in the Parameter Space of the Polynomials. The preliminary step in the curve tting approach for representing temporal data is the choice of the family of curves to approximate. Since we decided to use polynomials, the problem becomes the choice of curve degree. The lesser the degree, the smaller is the dimensionality of the parameter space and the faster is the tting algorithm. However, if the degree is too low, the approximation of the actual points of the series may be too loose and the method may fail to capture important properties of the data. Therefore one always needs to balance between these two important aspects.

In order to avoid arbitrariness at this stage, we adopted the following heuris-tic strategy. For each dataset, we rst selected randomly a small fraction of series and t polynomials with increasing degree. The ratio between the number of non zero coecients and the total number of coecients represents how complex the data are, and can be used to choose the proper degree for that specic dataset. We observed experimentally that straight lines are sucient if the level of noise is low (Fig. 1 left), while noisy data can be represented by means of second (at most third) degree curves (Fig. 1 right).

-Probability product kernels for HMM-based representation. The inizialisation stage is the very critical for the HMM-based method. First, we note that a major source of concern is the t of a multiparametric probabilistic model (such as the HMMs) to a single time series, and therefore the eectiveness of an abstract representation of two series in terms of probability distributions.

According to [START_REF] Jebara | Probability product kernels[END_REF], since the HMM is just an intermediate step in forming the kernel and to capture the similarity among series, it is not necessary to build a very accurate model. In principle, we agree with this viewpoint, but in our application scenario temporal series corresponding to the same behavioural pattern may be have very dierent lengths and structure due to errors in the tracking system. These may aect negatively the the algorithm that learns the HMM model.

In order to limit these concerns and reduce the eect of previous errors in the algorithmic pipeline, we test both a common initialisation for all the series and a fully independent model tting strategy. The results show that the original fully independent approach is not competitive in our scenario.

-P-spectrum kernels for string-based representations. When temporal series are described as strings over an alphabet, the rst issue is the choice of the alphabet symbols. Above, we already mentioned an automatic quantisation approach in which the feature space is partitioned by means of spectral clustering of all the points in the dataset. The association of a symbol/state to each point in the sequence is shown in Figure3(b), where the colours represent dierent symbols and each string is a concatenation of symbols. If the feature vectors are dened in a low dimensional space (d ≤ 3) -which may be the case in our context -then a viable alternative to automatic selection could be manual partitioning -where a user decides how to cluster reasonably the feature space (g. 3(a)).

For a quantitative assessment of the quality of the alphabet denition, we evaluated the stability of the representation in terms of how similar the corresponding strings are when two similar sequences are considered. Since the series in dataset DS3B and DS6B are known to belong to a xed set of behavioural classes, we evaluated the intra-class stability of the dierent methods. Similarity between strings is computed with the 2-spectrum. Figs. the xed manual partition, this approach is not suitable for real applications.

Experiments on temporal series categorisation

For each representation scheme, we computed the corresponding feature map, built the similarity matrix of all the series in the dataset and run the spectral clustering algorithm to evaluate the results.

As a rst analysis, we inspected the similarity matrices obtained with each kernel function and noise level, see Figure 4. In order to highlight the block structure of the matrices and make the non-quantitative analysis easier, we ordered the series beforehand insofar that the optimal method should produce exactly three distinct blocks along the diagonal of the similarity matrix. 2-spectrum kernel combined with string based-representation is very close to get this optimal result when the noise level is low, and it is also the closest one on more noisy data.

These results were conrmed also in the subsequent quantitative analysis.

In general, quantitative performance evaluation on the unsupervised domain is dicult to achieve, and no well assessed -and theoretically ground -methods have been proposed so far. The most popular solution to this problem is to analyse labelled datasets for which the ground truth is known -i.e. to use the clustering method as a classication algorithm -and to measure the classica-Fig. 4. Similarity matrices: from left to right, gaussian kernel on polynomial coecients, probabilistic product kernel on HMMs and 2-spectrum kernel on strings. The top row refers to moderately noisy data (0.3), the bottom row the noise level is increased (0.9). [START_REF] Camastra | A novel kernel method for clustering[END_REF].

In our experiments, we rely on the circumstance that for synthetic data it is possible to know the a-priori model by means of which they are generated.

Therefore, we can measure the stability of intra-class and inter-class distances, and compare the performances of the methods. Table 1 reports a comparative analysis of the results obtained as the noise level grows. The performances are expressed in percentages and refer to the correct classication of the clusters.

The above results provide experimental evidence that, in general, the approach based on the use of string kernel gives better results than the others.

Manual partition seems to be more accurate than the automatic method for As anticipated, in this specic experimental scenario, HMM-based technique depends strongly on the initialization strategy of the HMM density estimation: an independent inizialisation tends to map series of the same cluster to dissimilar feature vectors, and therefore the spectral clustering algorithm is not able to cut the graph appropriately. Furthermore, with the probability product kernel seem to fail to capture correctly the directionality of the series, if compared with the performances of 2-spectrum.

Polynomials give very poor results for higher levels of noise and do not consent to distinguish among bidirectional events: in curve tting we disregard the order in which the points are generated and, thus the direction of the sequence.

Discussion

Automatic behaviour understanding may be eectively addressed by unsupervised learning on temporal series. To our knowledge the available literature is lacking both on theoretical and experimental view-points. This paper proposes an analysis of a selection of methods to represent temporal series and analyse them via spectral clustering. All these methods can be cast in a common algorithmic scheme, although they highlight dierent aspects of the data at hand.

We evaluated the dierent choices on a set of synthetically generated data at dierent noise levels. This analysis is motivated by the intent of obtaining a robust method for temporal series analysis to be applied in the videosurveillance domain. Preliminary results, obtained on the PETS06 dataset, available at http://www.cvg.rdg.ac.uk/PETS2006/data.html, are promising. Trajectories have been obtained with a change detection module, followed by a Kalman tracker. A colour-based data association limits the eects of occlusions.

Trajectories model the person's centroid position evolving in time. Figure 5 shows the results obtained using a state-based representations with 2-spectrum kernel overlapping sets of trajectories with dierent colors describe bidirectional patterns.

Future works will be devoted to exploit the eects of long observations on a more complex environment and explore the eects of adopting more complete feature maps and kernels.

3 . 1 Fig. 1 .

 311 Fig. 1. Examples of series in the DS3B dataset. On the left the noise level is low and the trajectories are smooth and almost linear. On the right the noise level increases and the classication of the series is far more dicult.

3

  (c), 3(d) and 3(e) show mean and variance for dierent datasets arranged with increasing noise levels over the 20 replicates of the entire dataset. Blue plot refers to the automatic partition, red plot to the denition of a xed manual partition specied -and kept xed -at the beginning of all the experiments. Finally, in the green plot an ad-hoc partition is dened for each experiment: even if the results are slightly better that those obtained with

Fig. 3 .

 3 Fig. 3. The choice of a proper alphabet. The top-left gure shows the states assigned by manual choice, compared with the result of our automatic procedures (top-right). The botton row shows the stability of the representations while the noise level increases.

Fig. 5 .

 5 Fig. 5. Preliminary results on real data extracted from the dataset PETS06.

Table 1 .

 1 Percentages of correct behaviour associations on datasets DS3B and DS6B.

	Dataset DS3B	Dataset DS6B
	Noise Level 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9
	Strings (autom. alphabet) 90.0 80.7 80.0 81.0 79.0 97.8 96.2 87.0 84.1 78.8
	Strings (manual alphabet) 92.8 94.0 79.0 79.0 78.0 99.8 98.6 96.2 84.0 67.4
	HMMs (common init.) 87.7 81.5 70.4 63.1 58.8 48.2 44.6 39.4 35.4 31.7
	HMMs (indep. init.) 60.0 57.4 51.8 48.6 46.8 36.2 34.8 34.2 27.9 30.6
	Polynomials 91.0 72.0 56.0 40.0 43.0 -	-	-	-	-
	Normalised polynomials 62.0 57.2 52.0 50.5 47.1 -	-	-	-	-
	tion performances. Indeed, it is worth pointing our that in many applications
	of unsupervised learning one is not necessarily interested in assigning a class
	to each datum but to cluster coherent data, discarding noisy or uncommon
	patterns, see for instance