M Grott

J Knollenberg

M Hamm

K Ogawa

R Jaumann

K Otto

M Delbò

Patrick Michel

J Biele

W Neumann

Low thermal conductivity boulder with high porosity identified on C-type asteroid (162173) Ryugu

Low thermal conductivity boulder with high porosity identified on C-type asteroid (162173) Ryugu

M. Grott, J. Knollenberg, M. Hamm, K. Ogawa, R. Jaumann, K. Otto, M. Delbo, P. Michel, J.

Biele, W. Neumann, M. Knapmeyer, E. Kührt, H. Senshu, T. Okada, J. Helbert, A. Maturilli, N. of sand-to pebble-sized particles was found to be absent [START_REF] Jaumann | In-situ investigation of asteroid (162173) Ryugu by the Mobile Asteroid Surface Scout (MASCOT) Camera (MASCam)[END_REF][START_REF] Sugita | The geomorphology, color, and thermal properties of Ryugu: Implications for parent-body processes[END_REF] . Rather, the surface is largely covered by cobbles and boulders, seemingly incompatible with the infrared observations.

Here we report on the first in-situ thermal infrared observations of a boulder on a C-type asteroid. Boulder thermal inertia was found to be much lower than anticipated based on laboratory measurements of meteorites, which implies that the boulder itself exhibits low thermal conductivity. Our results furthermore indicate high boulder porosities as well as a low tensile strength in the few hundred kPa range. This confirms the suspected observational bias in our meteorite collections, as typical samples of common asteroid types would be too frail to survive atmospheric entry.

On October 3rd, 2018, the Hayabusa2 spacecraft [START_REF] Watanabe | Hayabusa2 Mission Overview[END_REF] delivered the MASCOT [START_REF] Ho | MASCOT-The Mobile Asteroid Surface Scout Onboard the Hayabusa2 Mission[END_REF] lander to the surface of asteroid (162173) Ryugu, where MASCOT's infrared radiometer MARA [START_REF] Grott | The MASCOT Radiometer MARA for the Hayabusa 2 Mission[END_REF] obtained surface brightness temperature measurements for a full day-night cycle. The instrument's field of view covered a spot of approximately 10 cm in diameter on the surface, which was also observed by MASCOT's optical camera MasCam [START_REF] Jaumann | In-situ investigation of asteroid (162173) Ryugu by the Mobile Asteroid Surface Scout (MASCOT) Camera (MASCam)[END_REF] . The scene observed by MARA and MasCam 4 is shown in Figure 1, with the footprint of the MARA 8-12 µm broadband sensor indicated in red. MARA is viewing the top face of a slightly raised (~3 cm) boulder in front of MASCOT. The boulder is rough in surface texture at the sub-centimeter scale and generally appears angular to sub-angular.

Particles with diameters larger than 0.6 mm are resolvable at the bottom of the image (~0.2 mm per pixel resolution), but only a few separate roundish grains can be identified as loose grains on this formation. Furthermore, an optically thick cover of fine particles is not observed, and bright inclusions are visible at multiple locations including the area viewed by MARA. Highly resolved areas in the MasCam image show that the boulder is composed of a relatively dark matrix with mostly bright but also darker inclusions at the millimeter scale [START_REF] Jaumann | In-situ investigation of asteroid (162173) Ryugu by the Mobile Asteroid Surface Scout (MASCOT) Camera (MASCam)[END_REF] .

Remote spectral observations of Ryugu suggest a composition similar to heated CM (CM2) or CI meteorites [START_REF] Moskovitz | Rotational characterization of Hayabusa II target Asteroid (162173)[END_REF] . However, the texture observed in close-up images (compare Fig. 1) shows millimeter-sized inclusions in at least one location and bright speckles at varying distances. This seems to be incompatible with a CM2 composition, for which petrographic analysis [START_REF] King | Grain Size and Petrography of C2 and C3 Carbonaceous Chondrites[END_REF] shows maximum grain sizes around 0.2 mm. On the other hand, CI meteorites are predominantly composed of a fine-grained matrix with a mottled appearance, and the texture observed by MASCOT resembles that of the rare CI2 Tagish Lake meteorite, which possesses up to 2 mm large calcium aluminum rich inclusions [START_REF] Brown | The Fall, Recovery, Orbit, and Composition of the Tagish Lake Meteorite: A New Type of Carbonaceous Chondrite[END_REF] . Thus, CI2 type meteorites appear to be the closest known Ryugu analogue. However, as Tagish Lake is a rare sample and variations in inclusion sizes are common for carbonaceous chondrites, it is not feasible to derive a definite meteorite analogue. Rather, it may be possible that the investigated boulder is not represented in any meteorite collection on Earth. As daytime data is affected by surface roughness and re-radiation from the environment, only equilibrated nighttime temperatures are used to fit thermophysical models. Best-fitting models (see methods) are indicated in Figure 2a and correspond to a thermal inertia Γ of 282 J m -2 K -1 s - 1/2 . While fitting nighttime data perfectly, the steep increase of temperature during the morning is underestimated, while midday temperatures are overestimated. Taking surface roughness into account (green solid line in Figure 2a), the quality of the fit to midday temperatures is much improved, but early morning temperatures cannot be fit using this model. The latter are influenced by light reflected from the MASCOT lander, and a local terrain model including reflections would be necessary to improve results. Nevertheless, since only equilibrium nighttime temperatures are used to estimate the thermal inertia Γ, presented results are largely independent of surface roughness and topography.

Admissible thermal inertia values are shown as a function of maximum insolation in Figure 2b, where the orientation of the surface normal has been systematically varied around its nominal value. The color bar shows the χ 2 of the individual fits, and emissivity has been varied between 0.9 and 1. Low Γ corresponds to low emissivity and models with and without re-radiation have been considered. As a result, admissible thermal inertia values for the boulder in the MARA field of view were found to be 282 J m -2 K -1 s -1/2 . This estimate is similar to telescopically determined thermal inertia values 1 of 150 to 300 J m -2 K -1 s -1/2 and values determined by the thermal infrared imager on the Hayabusa2 spacecraft [START_REF] Sugita | The geomorphology, color, and thermal properties of Ryugu: Implications for parent-body processes[END_REF] , which range from 200 to 500 J m -2 K -1 s -1/2 . It therefore seems likely that boulders dominate the thermal emission from Ryugu, which would be consistent with the high rock abundance determined from orbiter images [START_REF] Sugita | The geomorphology, color, and thermal properties of Ryugu: Implications for parent-body processes[END_REF] . Therefore, contrary to expectation, the low thermal inertia derived for Ryugu does not correspond to a pebble-sized regolith-covered surface [START_REF] Wada | Asteroid Ryugu Before the Hayabusa2 Encounter[END_REF][START_REF] Gundlach | A new method to determine the grain size of planetary regolith[END_REF] . Rather, boulder to block-sized clasts themselves appear to have thermal inertia lower than that of CM2 Cold Bokkeveld, which has the lowest thermal inertia (600 -700 J m -2 K -1 s -1/2 ) of a meteorite measured so far.

While thin layers of fine material could in principle mask the thermal signature of competent rock, the boulder observed by MARA appears to be free from an optically thick, dusty layer.

Furthermore, the presence of a fine dust layer can be ruled out by considering a two layer thermal model. Results of the calculations (see methods) are shown in Figure 3, where dust with a thermal inertia of 25 J m -2 K -1 s -1/2 was assumed to cover a boulder with a thermal inertia of 700 J m -2 K -1 s -1/2 . As is evident, the model is incompatible with the observed nighttime cooling rates, and the boulder observed by MARA itself must exhibit very low bulk thermal conductivity to fit the data. However, it cannot be ruled out that the low conductivity zone is limited to a highly porous outer layer. Such a layer may for example be generated by cracking due to thermal fatigue [START_REF] Delbo | Thermal fatigue as the origin of regolith on small asteroids[END_REF][START_REF] Hamm | Latitudinal Dependence of Asteroid Regolith Formation by Thermal Fatigue[END_REF][START_REF] Molaro | Thermally induced stresses in boulders on airless body surfaces, and implications for rock breakdown[END_REF] and could extend to a few thermal skin depths.

Given the above estimate of thermal inertia, thermal conductivity can be derived for a given bulk density and heat capacity (see methods). Furthermore, since thermal conductivity k of porous material depends more on porosity φ, and thus the total area of inter-grain contacts, than on the conductivity of the grains themselves 3 , the porosity of the investigated boulder could in principle be estimated if the functional dependence of k(φ) were known. However, while models for H and L chondrites indicate that porosity and thus thermal conductivity is governed by dehydration-or shock-induced cracks [START_REF] Krause | Thermal conductivity measurements of porous dust aggregates: I. Technique, model and first results[END_REF][START_REF] Henke | Thermal evolution and sintering of chondritic planetesimals III. Modelling the heat conductivity of porous chondrite material[END_REF][START_REF] Flynn | Physical properties of the stone meteorites: Implications for the properties of their parent bodies[END_REF] , thermal conductivity data on CI chondrites are absent. Nevertheless, since cracks are abundant in CI types [START_REF] Tonui | chemical and spectroscopic evidence for thermal metamorphism in carbonaceous chondrites I: CI and CM chondrites[END_REF] , it stands to reason that a similar mechanism would reduce thermal conductivity in these types and models derived for H and L chondrites should be applicable to CI types, too.

Two models [START_REF] Henke | Thermal evolution and sintering of chondritic planetesimals III. Modelling the heat conductivity of porous chondrite material[END_REF][START_REF] Flynn | Physical properties of the stone meteorites: Implications for the properties of their parent bodies[END_REF] for thermal conductivity as a function of porosity are shown in Figure 4a.

Available data for H, L, and CM chondrites are shown together with the thermal conductivities extrapolated for Ryugu. The first model [START_REF] Henke | Thermal evolution and sintering of chondritic planetesimals III. Modelling the heat conductivity of porous chondrite material[END_REF] results in 28 < φ < 34%, typical for CM and CI chondrites, which have class average porosities of 22.2% and 34.9%, respectively [START_REF] Flynn | Physical properties of the stone meteorites: Implications for the properties of their parent bodies[END_REF] . The second model 17 yields 41 < φ < 55%. Other models applicable to partially-sintered granular material [START_REF] Henke | Thermal evolution and sintering of chondritic planetesimals III. Modelling the heat conductivity of porous chondrite material[END_REF] yield intermediate results. Corresponding thermal conductivities are 0.06 -0.13 and 0.09 -0.16 W m -1 K -1 , respectively, and thus much lower than measurements reported for the thermal conductivities of meteorites. Measurements on CI chondrites are missing entirely, but since CI chondrites are those meteorites with highest porosities [START_REF] Flynn | Physical properties of the stone meteorites: Implications for the properties of their parent bodies[END_REF] , low conductivities could be expected.

Pores in the boulder would be observable in nighttime images, as the illumination provided by the MasCAM LED array covers a variety of directions. However, no such pores are observed, indicating that potential pores must be smaller than 1 mm. The radiative contribution to the total heat transport inside the boulder can now be estimated by considering radiative heat exchange between parallel planes 3 , and for pores smaller than 1 mm this results in a minor contribution to the overall thermal conductivity. The latter must therefore be governed by solid conduction through grain contacts. Given porosity and thermal conductivity, the amount of contacts can be estimated, which in turn can be converted to an estimate of tensile strength (see methods). Given the values derived above and assuming a Young's modulus representative for carbonaceous chondrites, tensile strength of the boulder is estimated to be 200 to 280 kPa and thus considerably lower than measurements on meteorite samples [START_REF] Flynn | Physical properties of the stone meteorites: Implications for the properties of their parent bodies[END_REF] , which generally show tensile strengths of the order of one to a few MPa [START_REF] Ostrowski | The physical properties of meteorites[END_REF] . This low tensile strength indicates an observational bias, namely that any hypothetical meteoroid originating from the boulder observed by MASCOT would likely break up during atmospheric entry and would thus be absent in our meteorite collections.

Thermal conductivity values derived above can be compared to estimates for other minor bodies, and corresponding data are shown in Figure 4b, where thermal conductivity is given as a function of temperature for asteroids (162173) Ryugu The high porosities derived here have important implications for Ryugu's parent body.

Assuming initial porosities 27 of the order of 70%, the Ryugu precursor needs to have been large enough to reduce intrinsic (micro-) porosity to values below 55% by compaction and aqueous alteration, while simultaneously avoiding porosities to drop below 28%. This implies that the material we observe on Ryugu's surface today was either produced in the outer layers of a larger (50-100 km sized) parent body, or the interior of a smaller, kilometer-sized body. The latter would be feasible provided accretion occurred while 26 Al was still active, in which case aqueous alteration and hot pressing of the precursor material would be efficient. In this way, porosities of 45% can be achieved starting from initial porosities of 70% even on small, kilometer-sized objects, provided that a water ice dominated primordial composition similar to that of CI/CM chondrites is assumed Correspondence and requests for materials should be addressed to Matthias Grott (Matthias.Grott@dlr.de) Results for four different dust thicknesses assuming a boulder thermal inertia of 700 J m -2 K -1 s - 1/2 are shown.

Methods

Asteroid Thermophysical Model. To estimate thermophysical properties from the brightness temperatures observed, a thermophysical model of the Ryugu boulder was constructed assuming a single, flat surface in the MARA field of view. A specific heat c p of 600 J kg -1 K -1 and density ρ of 1270 kg m -3 have been assumed for the boulder, consistent with pre-encounter estimates [START_REF] Wada | Asteroid Ryugu Before the Hayabusa2 Encounter[END_REF] .

Parameters have been assumed to be constant and independent of temperature, as more complicated models did not improve the quality of the fit. Thermal conductivity k was treated as a free fitting parameter, and results are reported in terms of surface thermal inertia =

Insolation of the surface in the field of view was varied around the average surface normal which, according to the Ryugu shape model, points towards longitude 314.207°E and latitude 34.599°S in the asteroid fixed frame at the landing site. In a local frame, elevation and azimuth of the normal vector have been varied by 25° and 360° around this surface normal, respectively.

This accounts for the unknown orientation of the surface in the MARA field of view and covers all plausible illumination conditions. Results are reported in terms of the maximum insolation corresponding to the respective surface normal. Furthermore, the times of sunrise and sunset have been adapted to fit MARA observations, which indicate that sunrise is delayed by 37 min with respect to the nominal insolation. In addition, the sun sets 21 min earlier than predicted by the illumination model and insolation has been adapted accordingly. This is consistent with the terrain at the landing site, which shows that MASCOT is situated in a local depression.

Re-radiation from the environment onto the surface observed by MARA was taken into account using a local terrain model. View factors f from the surrounding topography to the surface have been estimated from the model and were found to amount to 0.048 ± 0.007 for facets within 1 meter of the MASCOT landing site (see supplementary Figures 1 and2). No facet showed > 0.08. In the thermophysical modeling, f has been varied between 0 and 0.08 to provide a conservative upper limit. The incident flux was then derived assuming that the surrounding terrain has temperatures corresponding to those on the surface of the boulder observed by MARA. As thermal re-radiation decreases the estimated thermal inertia, f = 0.08 results in an upper limit for the derived thermal inertia.

Given insolation and thermal inertia Γ, the 1D heat equation is solved 30 using

-Γ = (1 -) - + (2)
as the upper boundary condition. Here, is surface temperature, Γ thermal inertia, rotation period, bond albedo, the Stephan-Boltzmann constant, depth normalized to the diurnal skin depth [START_REF] Spencer | Systematic biases in radiometric diameter determinations[END_REF] , emissivity, and the view factor to the surrounding environment. In the calculations, an albedo of = 0.0146 is assumed while emissivity was varied between 0.9 and 1. The rotation period of Ryugu is 7.6326 h.

Data Fitting. To avoid complications caused by inhomogeneous temperatures in the MARA field of view due to the changing illumination conditions and re-radiation from the surroundings, only equilibrated nighttime data were used to fit thermophysical models [START_REF] Christensen | The Thermal Emission Imaging System (THEMIS) for the Mars 2001 Odyssey Mission[END_REF] and invert for thermophysical parameters. To fit the data, a suite of models was computed by systematically varying thermal inertia and the orientation of the surface normal in a grid search approach. In the models, thermal inertia was varied in steps of 1 J m -2 s -1/2 K -1 between 170 and 410, elevation of the target surface normal was varied in steps of 5° between 90° and 65°, and azimuth of the target surface normal was varied in steps of 10° between 0 o and 360 o . For each model, the value between model and data was computed and models resulting in a larger than a critical value were discarded. Here we choose a critical corresponding to a 2 confidence interval 30

0.05 = ( , )
where ( , ) is the -distribution and is equal to the number of data points minus the number of fitting parameters. We obtain a critical of 587.8 for the 534 data points obtained between 17:15 and 07:16 local time used in the fitting. The minimum obtained for the best fit was found to be = 66.

This procedure was repeated by varying surface emissivity and assuming values of ε = 0.9, 0.95 and 1. Furthermore, models with view factors to the surroundings of = 0 and 0.08 have been computed to study the effect of re-radiation, which was found to have only a small influence on nighttime temperatures. The stated uncertainty for the thermal inertia was then derived from the lowest and highest admissible values of all simulations. The best-fitting model has an emissivity of ε = 1. For consistency, the emissivity used for both the thermophysical models and the derivation of surface temperature from the observed flux was identical in all cases.

Surface Roughness. For a rough surface, observed flux depends on the solar zenith angle φ, the angle between surface normal and observation direction δ, as well as the angle between projections of the sun vector and the observation direction onto the surface ψ. These need to be taken into account when calculating fluxes received by the instrument. Here, the rough surface is modeled as a surface covered by spherical-section craters [START_REF] Kuehrt | Interpretation of the KRFM-Infrared Measurements on Phobos[END_REF][START_REF] Giese | Theoretical Interpretation of Infrared Measurements at Deimos in the Framework of Crater Radiation[END_REF] , assuming lateral heat transport to be negligible. Using this model, a factor ( , , , , ) is calculated for each time t and observed wavelength relating the flux emitted by a flat surface to the flux emitted by the rough surface = ( , , , , ) . This model is scale-independent [START_REF] Kuehrt | Interpretation of the KRFM-Infrared Measurements on Phobos[END_REF] , leaving the crater density on the surface and the opening angle of the craters as free parameters. The latter is kept constant at 180°, corresponding to a surface covered by hemispherical depressions. The model also depends on parameters such as albedo and heliocentric distance, which are all kept constant to be consistent with the thermophysical model described above. was calculated at = 10 .

Two Layer Model. To estimate the possible influence of a low conductivity dust layer on the modeling results, a two layer regolith model assuming geometry parameters corresponding to those of the best fitting one-layer model was calculated. In the model, the dust layer is treated in the continuum approximation, i.e., it is assumed that thermal transport properties inside the layer can be treated using the theory of porous media [START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF] . This approximation breaks down if thin layers of dust are present, in which case radiative heat transport inside the dust layer would need to be Impact experiment on 9P/Tempel 1. For the heat capacity, a value of 1000 J kg -1 K -1 was used 23 .

For the computation of thermal conductivities for (101955) Bennu, (25143) Itokawa, and average NEA, heat capacity 1 was evaluated at a temperature of 340 K, while for the disc-averaged measurement of (162173) Ryugu a temperature of 277 K has been assumed. Strength Estimate. To estimate mechanical tensile strength given thermal conductivity k and porosity φ, the amount of inter-grain contacts is estimated neglecting radiative heat transport

using 3 = 4 (1 -) (6)
where the grain thermal conductivity k s has been assumed to be 2.95 W m -1 K -1 as appropriate for serpentine. Here, H = r c /R is the Hertz factor with r c being the contact radius and R the radius of interacting spheres. For the coordination number C, the relation

= 2.812(1 -) / (1 + ) (7)
was used, where = 0.07318 + 2.193 -3.357 + 3.194

Given k, k s , φ, C and H can then be calculated. For a random packing of identical, isotropic and homogeneous spheres with shear modulus and Poisson's ratio , the bulk elastic parameters can then be related to those in a porous medium following the approach of Digby 36 : in Hertzian contact theory, two identical spheres have a circular contact area of radius , depending on the confining pressure and the Young's modulus of the sphere material. In extension of the Hertz contact model, Digby assumes that the contact area between two spheres contains a small, concentric area of radius < where the spheres are firmly bonded, i.e., remain in contact even without confining pressure. He obtains the effective Lamé parameters of the packing

= (1 -) 5 1 - - 2 2 - (9) = 1 2 (1 -) 5 2 1 - + 6 2 - (10)
We take the adhesive region as a model for a sintering neck between two particles and consider the case without confining pressure, where ≡ . Using the relation

= 3 + 2 + ( 11
)
between Lamé-parameters and Young's modulus, and the relation

= 2 (1 + ) (12)
between shear modulus, Lamé parameter and Young's modulus , we obtain the effective Young's modulus

1 , 2 , 3 .

 123 Müller, A. Hagermann, N. Sakatani, S. Tanaka, T. Arai, S. Mottola, S. Tachibana, I. Pelivan, L. Drube, J.-B. Vincent, H. Yano, C. Pilorget, K. D. Matz, N. Schmitz, A. Koncz, S. Schröder, F. Trauthan, M. Schlotterer, C. Krause, T.-M. Ho, A. Moussi-Soffys C-type asteroids are among the most pristine objects in the solar system, but little is known about their interior structure and surface properties. Telescopic thermal infrared observations have so far been interpreted in terms of a regolith covered surface with low thermal conductivity and particle sizes in the centimeter range. This includes observations of C-type asteroid (162173) Ryugu, for which average grainsizes of 3-30 mm have been derived However, upon arrival of the Hayabusa2 spacecraft at Ryugu, a regolith cover

  Figure 2a along with the best-fitting thermal models taking the illumination conditions at the landing site as well as surface roughness into account (see methods). Diurnal temperatures rise steeply in the morning, but start dropping around 11:07 local time, indicating shadows passing through the radiometer field of view before noon. Maximum temperatures reach 308 K shortly after local noon and the sun sets at 16:39 local time. The complex shape of the daytime temperature curve indicates a rough surface, consistent with camera observations. During nighttime, temperatures drop to 201 K.

Figure 1 :

 1 Figure 1: MasCam Image of the boulder observed by MASCOT indicating the MARA field of view (red shaded area). a) The location in daylight (local time 9:20) with the yellow arrow indicating the approximate direction of illumination with sun elevation and azimuth at 40.2° and 67.2°. The image suggests that MASCOT is located in front of an angular to sub-angular formation whose edges are outlined by the yellow dotted line. The yellow dashed line indicates the edge of an elevated part of the boulder (compare the nighttime image on the right). The front face of the formation orientated towards MARA is approximately 3 cm above the plane MASCOT is located on. b) The same location at night (local time 23:18) illuminated by the camera's red LED. Only the foreground is visible due to the limited illumination provided. The images are distorted with pixel resolutions varying between approximately 0.2 mm at the bottom

Figure 2 :

 2 Figure 2: Observed and modeled surface temperatures and derived thermal inertia. a) Variation of surface temperature observed in-situ at geographical coordinates of 22.22 ± 0.05°S, 317.26 ± 0.07°E. Temperature is shown as a function of time as derived from the MARA 8-12 µm filter and using a surface emissivity of ε = 1. Error bars indicate 2σ confidence limits and uncertainties are below 0.5 K and 1.5 K during daytime and nighttime, respectively. Data are shown together with best fitting thermal models. While the model shown as a dashed red line assumes a flat surface, midday temperatures are reduced by surface roughness effects for the model shown in green. The steep rise of morning temperatures is caused by reflections from theMASCOT lander (not modeled). Best fitting models correspond to a thermal inertia of 282 J m -2 K -1 s -1/2 and a crater density of 0.34 (see methods). b) Retrieved thermal inertia as a function of the maximum insolation for the respective surface orientation. Emissivity has been varied between ε = 0.9 and 1. The χ 2 value of the individual fits is shown in color and admissible thermal inertia values were found to be 282 J m -2 K -1 s -1/2 .

Figure 3 :

 3 Figure 3: Modeled temperatures for a dust covered surface. Variation of surface temperature as a function of time as derived from the MARA 8-12 µm filter and using a surface emissivity of ε = 1. Error bars indicate 2σ confidence limits and uncertainties are below 0.5 K and 1.5 Kduring daytime and nighttime, respectively. Data are shown together with the results of a two layer thermal model, which assumes best fitting illumination conditions as well as a thin dust layer covering the underlying boulder. Dust thermal inertia is assumed to be 25 J m -2 K -1 s -1/2 .

Figure 4 :

 4 Figure 4: Derived thermal conductivity and boulder porosity. a) Thermal conductivity as a function of porosity. Data for H, L, and CM chondrites is shown together with two models fitting the data for porosities below 20%. For the low thermal conductivities determined from MARA measurements, models have to be extrapolated to high porosities. Depending on the model used, porosities between 28-34% and 41-55% are obtained. b) Thermal conductivity as a function of temperature as derived for different small bodies. For the boulder in the MARA field of view, a

  Assumed bulk densities for (162173) Ryugu 1 , average NEA 21 , (25143) Itokawa 25 , and (101955) Bennu were 1270 kg m -3 , 1950 kg m -3 , 1950 kg m -3 , 1270 kg m -3 , respectively.

  of sintered spheres in relation to the Young's modulus of the matrix material. Since 0 ≤ ≤ 1 2 ⁄ usually holds for Poisson's ratio, the dependence of on amounts to a factor between 0.988 and 8 7 ⁄ , or to 256 255 ≈ 1.004 ⁄ if the matrix material isPoissonian. Therefore, we finally obtain (reduced) Young's modulus is then converted into an estimate of tensile strength σ t using the empirical relation[START_REF] Scholz | The Mechanics of Earthquakes and Faulting, 2 nd Edition[END_REF]

  [START_REF] Wada | Asteroid Ryugu Before the Hayabusa2 Encounter[END_REF] , (101955) Bennu[START_REF] Dellagiustina | Properties of rubble-pile asteroid (101955) Bennu from OSIRIS-REx imaging and thermal analysis[END_REF] , and the average estimated for small near Earth asteroids 21 . Values are compatible within error bars and similar to values derived for comets 67P/Churyumov-Gerasimenko 22 , 9P/Tempel 123,24 , and 103P/Hartley 2 23 when heat capacity is properly scaled for temperature and appropriate densities are assumed

(see methods section). On the other hand, estimates for S-type asteroid (25143) Itokawa 25,26 are larger by a factor of three, indicating that thermal properties of C-type asteroids are more similar to those of comets than those of S-type asteroids. It is also worth noting that no cubic dependence of thermal conductivity on temperature is observed in Figure

4b

, which could indicate that radiative heat transfer through large pores is negligible in the bodies considered here.

  28,29 . 21. Delbo, M., dell'Oro, A., Harris, A.W., Mottola, S., Mueller, M. ,Thermal inertia of near-Earth asteroids and implications for the magnitude of the Yarkovsky effect, Icarus 190, 236-

	249 (2007).
	22. Spohn, T., Knollenberg, J., Ball, A.J., et al., Thermal and mechanical properties of the near-
	surface layers of comet 67P/Churyumov-Gerasimenko, Science 349: 6247 (2015).
	23. Groussin, O. et al., The temperature, thermal inertia, roughness and color of the nuclei of
	Comets	103P/Hartley	2	and	9P/Tempel,	Icarus	222,	580-594	doi:
	10.1016/j.icarus.2012.10.003 (2013).
	24. Davidsson, B. J. R., Gutiérrez, P. J., Rickmann, H., Physical properties of morphological
	units on comet 9P/Tempel 1 derived from near-IR Deep Impact spectra, Icarus 201, 335-357
	doi: 10.1016/j.icarus.2008.12.039 (2009).
	Astron. Astrophys., 584, A117, doi:10.1051/0004-6361/201527083 (2015).

25. Fujiwara, A., Kawaguchi, J., Yeomans, D.K., Abe, M., et al., The Rubble-Pile Asteroid Itokawa as Observed by Hayabusa, Science 312: 1330-1334 (2006). 26. Müller, T.G., Hasegawa, S., Usui, F., (25143) Itokawa: The power of radiometric techniques for the interpretation of remote thermal observations in the light of the Hayabusa rendezvous results, Publ. Astron. Soc. Japan 00 (0), 1-17, doi: 10.1093/pasj/psu034 (2014). 27. Yang, R.Y., Zou, R.P., Yu, A.B., Computer simulation of the packing of fine particles, Physical Review E, 62(3), 3900-3908. doi:10.1103/physreve.62.3900 (2000). 28. Neumann, W., Breuer, D., Spohn, T., On the modelling of compaction in planetesimals, Astr. Astrophys., 567, A120, doi:10.1051/0004-6361/201423648 (2014). 29. Neumann, W., Breuer, D., Spohn, T., Modelling the internal structure of Ceres: Coupling of accretion with compaction by creep and implications for the water-rock differentiation.

Acknowledgments

The MASCOT lander on the Hayabusa2 Mission of JAXA is a DLR/CNES cooperation.

MASCOT

MARA has been developed and built under the leadership of the DLR Institute of Planetary Research with contracted contributions from the Institute of Photonic Technology. The Hayabusa2 Mission is operated by JAXA. Funding: K.O. acknowledges funding by the JSPS Core-to-Core Program "International Network of Planetary Sciences." A.H. acknowledges funding by STFC under grant no. ST/S001271/1.

The Young's modulus E of ordinary chondrites is usually reported 17 to be of the order of tens of GPa, while values for carbonaceous chondrites are usually much lower 38,39 , ranging from a few to about 10 GPa. Here we assume 10 GPa to be representative for the Ryugu boulder, pointing out that this likely places an upper limit on the derived tensile strength.

Data Availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Author contributions

M.G. coordinated and wrote the paper, J.K. and M.G. evaluated the instrument calibration, M.H., K.O., S.M., J.-B.V., M.S. and I.P. computed illumination and thermal models, N.S., S.S., A.K. Here we assume that the dust layer has a thermal inertia of 25 J m -2 K -1 s -1/2 . This corresponds to typical particle sizes of ~10 µm at porosities of 80% [START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF] . Therefore, our model assumptions are valid for layer thicknesses in excess of 50 µm.

Models have then been calculated varying the thermal inertia of the bottom layer along with the top layer thickness. Even thin dust layers appreciably distort the temperature curves, and using bottom layer thermal inertias between 250 and 700 J m -2 K -1 s -1/2 , nighttime cooling curves could only be fitted within the errorbars for thermal inertias close to the values reported above. This indicates that the thermophysical properties of the boulder itself govern thermal emission.

Thermal Conductivity and Porosity

for a given functional dependence for ( ). Here, Γ is thermal inertia, c p is specific heat, ρ s is grain density, and φ is porosity. For the grain density ρ s , the average density of CI chondrites [START_REF] Flynn | Physical properties of the stone meteorites: Implications for the properties of their parent bodies[END_REF][START_REF] Macke | Density, porosity, and magnetic susceptibility of carbonaceous chondrites[END_REF] ρ s = 2420 kg m -3 has been assumed. Heat capacity was determined by evaluating 1 c p (T) at T = 230 K. For the functional dependence of ( ), two models have been considered. Both are based on empirical fits to available thermal conductivity data for chondritic meteorites, and the first relation [START_REF] Flynn | Physical properties of the stone meteorites: Implications for the properties of their parent bodies[END_REF] is

while the second relation [START_REF] Henke | Thermal evolution and sintering of chondritic planetesimals III. Modelling the heat conductivity of porous chondrite material[END_REF] is

(

Both formulations fit the experimental data equally well, but diverge at porosities larger than 20%.

To compare the thermal conductivity estimates for meteorites with those for comets, cometary thermal inertia estimates have been converted to an estimate of thermal conductivity assuming a cometary bulk density of 300 kg m -3 as derived from spacecraft observations 23


Etienne Fodor

Vishwajeet Mehandia

Jordi Comelles

Raghavan Thiagarajan

Nir S Gov

Paolo Visco

Frédéric Van Wijland

Daniel Riveline

É Fodor

Spatial Fluctuations at Vertices of Epithelial Layers: Quantification of Regulation by Rho Pathway

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Changes in shapes of cells and tissues are mediated by the acto-myosin cytoskeleton. To reproduce the dynamics of this network, minimal systems made of actin filaments, myosin motors and cross-linkers are synthetized in vitro [START_REF] Backouche | Active gels: dynamics of patterning and self-organization[END_REF][START_REF] Soares E Silva | Time-resolved microrheology of actively remodeling actomyosin networks[END_REF][START_REF] Blanchoin | Actin dynamics, architecture, and mechanics in cell motility[END_REF][START_REF] Murrell | Forcing cells into shape: the mechanics of actomyosin contractility[END_REF][START_REF] Schuppler | Boundaries steer the contraction of active gels[END_REF]. The mechanics and dynamics of such active gels are controlled by varying the concentration of their various components. Activity of each component is monitored by adding some inhibitor drugs, and/or by tuning the ATP concentration of the system. Recent experimental evidence have shown the relevance of this approach to investigate the role of motors and cross-linkers in the emerging properties of the network [START_REF] Soares E Silva | Time-resolved microrheology of actively remodeling actomyosin networks[END_REF][START_REF] Soares E Silva | Active multistage coarsening of actin networks driven by myosin motors[END_REF]. These studies are based on tracking the motion of tracers injected in active gels: analyzing the spontaneous fluctuations of such tracers enables one to extract information about the activity of internal motors.

In multicellular systems, such as tissues of developing embryos, acto-myosin drives the morphogenesis: dramatic rearrangements leading to the formation of distinct organs [START_REF] Rauzi | Nature and anisotropy of cortical forces orienting Drosophila tissue morphogenesis[END_REF][START_REF] Guillot | Mechanics of Epithelial Tissue Homeostasis and Morphogenesis[END_REF]. This remodelling is mainly under the control of intracellular activity, which powers spatial fluctuations [START_REF] Marmottant | The role of fluctuations and stress on the effective viscosity of cell aggregates[END_REF], and intercellular interactions mediated by adhesion between neighboring cells [START_REF] Cavey | A two-tiered mechanism for stabilization and immobilization of E-cadherin[END_REF][START_REF] Bardet | PTEN Controls Junction Lengthening and Stability during Cell Rearrangement in Epithelial Tissue[END_REF]. In contrast to synthetic gels, the internal regulation of the cellular acto-myosin activity is more complex in vivo. Therefore, extending the in vitro approach, based on controlling externally the activity of each specific component, to in vivo situations requires new strategies.

The Rho signaling pathway is known to regulate the actomyosin activity in living cells [START_REF] Etienne-Manneville | Rho GTPases in cell biology[END_REF]. It also controls cell-cell junctions [START_REF] Reyes | Anillin Regulates Cell-Cell Junction Integrity by Organizing Junctional Accumulation of Rho-GTP and Actomyosin[END_REF] and the elasticity of stress fibers [START_REF] Oakes | Optogenetic control of RhoA reveals zyxin-mediated elasticity of stress fibres[END_REF]. Such a pathway can be viewed as a series of activators and inhibitors installing a hierarchy of potential targets [START_REF] Gibson | The emergence of geometric order in proliferating metazoan epithelia[END_REF][START_REF] Riveline | Focal Contacts as Mechanosensors[END_REF]. Activations and inhibitions controlled by each target are such that anticipating their net effects on the tissue fluctuations, powered by acto-myosin activity, remains a challenge [START_REF] Besser | Coupling biochemistry and mechanics in cell adhesion: a model for inhomogeneous stress fiber contraction[END_REF][START_REF] Bement | Activatorinhibitor coupling between Rho signalling and actin assembly makes the cell cortex an excitable medium[END_REF]. In that respect, the inherent complexity of internal activity in vivo calls for new experiments and quantitative analysis to bridge the biochemical signaling of the Rho pathway with the emerging tissue dynamics.

In this paper, we explore the regulation of active fluctuations by the Rho pathway in epithelial monolayers. We measure these fluctuations by tracking tricellular junctions or vertices over time. In contrast with active gels, our analysis of internal fluctuations does not require to inject external tracers. Based on a phenomenological model, we quantify key parameters of junction activity: their effective diffusion coefficient, as well as the persistence time and persistence length of spatial fluctuations. We report modifications of these parameters for various targets along the signaling pathway. These results support that, for the inhibitions that we considered, the active fluctuations of the vertices are reduced when going downstream in the Rho pathway inhibition.

MATERIALS AND METHODS

Experiments were performed with MDCK II cells stably expressing E-cadherin GFP (Nelson Lab.). We culture cells in DMEM containing 10% Fetal Calf Serum (FCS) and antibiotics. We replate them on glass coverslips (CS) of 25 mm diameter for live cell imaging. When the cell monolayer covered 70% of the CS area, we firmly place the sample at the bottom of a custom made metallic holder. For acquisition, we change the medium to L15, 10% FCS and antibiotics. We use the following inhibitors from myosin up to Rho at optimal concentrations following the manufacturer recommendations: inhibition of acto-myosin by ML-7 (Sigma-Aldrich, 10 µM), inhibition of Rho kinase (ROCK) by Y-27632 (Sigma-Aldrich, 10 µM), and inhibition of Rho by C3 Transferase (Cytoskeleton, 0.04 µM). Note that the use of blebbistatin to inhibit directly myosins yielded some detrimental effects, leading us to rather use ML-7 instead.

For observation, we use a motorized inverted microscope (Nikon Eclipse Ti), equipped with a 12 bit CCD camera (Photometric CoolSNAP HQ2). The setup is temperature controlled at 37 • C (Life Imaging Services). We check with fluorescent beads (4 µm, TetraSpeck, Invitrogen Molecular Probes) grafted on CS surface that no drift appears during 24 hours of live imaging after 2 hour stabilization. We take pictures of the monolayer every 5 min during the next 8 hours with multiple z-stacks 1 µm apart. They span 3 µm depth of the cell monolayer. We merge the z-stacks into one image by using the maximum intensity projection. We then extract vertex positions from the sequence of merged images by manually clicking in each frame as long as they are visible. The procedure was validated in its precision through automatised detection as well. For each condition, we check that the average cell area was always about 180 ± 15 µm 2 , and we consider more than 20 vertices for at least 3 biological repeats.

RESULTS

Vertex tracking and inhibitors in the Rho pathway

We use Madin Darby Canine Kidney (MDCK) cells stably transfected with E-cadherin fused with the Green Fluorescent Protein (GFP) as a paradigm for epithelial tissues dynamics [START_REF] Adams | Mechanisms of Epithelial Cell-Cell Adhesion and Cell Compaction Revealed by High-resolution Tracking of E-Cadherin-Green Fluorescent Protein[END_REF]. This allows us to study live cells while interacting with each other. We seek to identify spatial points primarily involved in tissue transformations. The meeting points between three cells are involved in exchanges between neighbouring cells, thus serving as hallmark of tissue dynamics [START_REF] Bardet | PTEN Controls Junction Lengthening and Stability during Cell Rearrangement in Epithelial Tissue[END_REF][START_REF] Salomon | Contractile forces at tricellular contacts modulate epithelial organization and monolayer integrity[END_REF]. Some specific proteins, such as tricellulin, are known to accumulate at this point in cell culture. Besides, vertices have also attracted the attention of developmental biologists which exhibit accumulation of proteins at these specific points [START_REF] Lye | Subcellular localisations of the CPTI collection of YFP-tagged proteins in Drosophila embryos[END_REF].

Vertex dynamics are driven both by thermal fluctuations and by active fluctuations powered by some internal nonequilibrium processes such as motor-induced forces, actin polymerization and cell-cell adhesion. The myosin-II motors are localized in dense contractile units present in the apical surface of the tissue [START_REF] Rauzi | Planar polarized actomyosin contractile flows control epithelial junction remodelling[END_REF][START_REF] Klingner | RhoA and RhoC have distinct roles in migration and invasion by acting through different targets[END_REF] [Figs. 1(a-c)]. The active forces lead to large displacements of the vertices distinct from the thermal fluctuations of smaller amplitude. We focus here on large displacements which do not lead to any topological transitions in the tissue [Figs. 1(d-e)]. The Rho signaling pathway controls the acto-myosin activity inside cells, namely the forces induced by myosin and/or by actin polymerization (see Fig. 9 in ( 16)). In this study, we focus on the downstream targets that affect myosin to establish the principles for the validity and relevance of our framework. Upstream and downstream targets install a hierarchy in the activation of myosin. We specifically inhibit the following targets: Rho, Rho kinase (ROCK) and myosin-II [Fig. 1(f)]. Each inhibitor is specific to its target and incubated at the optimal concentrations for its inhibition. We used standard concentration values already utilized in other cell biology studies for many cell types including MDCK [START_REF] Jin | Myosin II Light Chain Phosphorylation Regulates Membrane Localization and Apoptotic Signaling of Tumor Necrosis Factor Receptor-1[END_REF][START_REF] Reffay | Interplay of RhoA and mechanical forces in collective cell migration driven by leader cells[END_REF][START_REF] Sorce | Mitotic cells contract actomyosin cortex and generate pressure to round against or escape epithelial confinement[END_REF][START_REF] Imai | Three-dimensional morphogenesis of MDCK cells induced by cellular contractile forces on a viscous substrate[END_REF]. Altogether, we probe four conditions on the same system by considering untreated cells as a control.

To demonstrate that the chosen inhibitors and their concentration specifically act on the contractile state of cells, we investigate their effect on both architecture and contractile forces within tissues by measuring single cell area and myosin cluster area in each condition. The distribution of single cell area is not strongly affected in the myosin-II inhibitor case compared with control, as apparent from the mean value in Fig. 1(g). The distribution gets modified in the Rho kinase and Rho inhibitor cases, with mean values being slightly reduced and increased, respectively. In contrast, the polygonicity distribution remains approximately the same for all conditions at different times, supporting that tissue architecture is barely affected by external inhibitors (see Fig. S1 in the Supporting Material).

Myosin clusters have been shown to be relevant readouts to assess the contractile state of cells [START_REF] Munjal | Actomyosin networks and tissue morphogenesis[END_REF][START_REF] Wollrab | Still and rotating myosin clusters determine cytokinetic ring constriction[END_REF]. Along this line, measurements of cluster characteristics are informative. In our case, the mean density value of myosin clusters for control is larger than in the inhibited cases (see Fig. S2 in the Supporting Material), suggesting a decrease in force generation for the same level of myosin per cell. In addition, the area of each myosin cluster is smaller in control than in other conditions [Fig. 1(h)]. Besides, this area increases the more downstream along the Rho pathway inhibition, suggesting a relaxation of the myosin pool in the apical side, consistently with the notion that myosin-induced forces are reduced. Altogether, our analysis of clusters confirms that we are acting on the contractile state of cells. Since inhibitors are specific and used at their optimal concentrations, these measurements support the validity of our experimental approach.

Statistics of vertex displacement: inhibitors affect spatial fluctuations

Our goal is to investigate how the emergent fluctuations of the tissue are regulated by the Rho pathway. To this aim, we first demonstrate that the inhibitors in the pathway affect these fluctuations by extracting the statistics of displacements from the vertex trajectories. This allows us to assess the existence of a direct link between biochemical signaling and mechanical fluctuations. In our analysis, we do not consider neither spatial inhomogeneities nor topological transitions that occur in the tissues. In that respect, we measure vertex trajectories in the absence of neighboring cell division, by tracking them as long as they are visible until a maximum of 8 hours.

We compute the projected one-dimensional mean square displacement (MSD) within the four different conditions [Fig. 2(a)]. For each condition, the short time MSD exhibits a power-law behavior with exponent close to 0.7 over about one decade. Interestingly, a subdiffusive behavior was reported for the dynamics of vertices in the endoplasmic reticulum [START_REF] Nixon-Abell | Increased spatiotemporal resolution reveals highly dynamic dense tubular matrices in the peripheral ER[END_REF]. The large time MSD depends on conditions and exhibits a behavior that, for simplicity, we have characterized by a power-law. The corresponding exponent is typically larger than 1, except for myosin inhibitor where it is smaller. The crossover between the two regimes appears between 20 and 60 min. Fluctuations are reduced in the myosin inhibited case, which has the lowest MSD, and they are enhanced for the Rho inhibitor, where the long time MSD is the largest. We also explore the full statistics of vertex displacement by measuring the probability distribution function (PDF) for each condition, as shown in Figs. 2(b-e). At short time the PDF is Gaussian, while it exhibits broader tails at large time. These tails reveal large displacements of the vertex, and were already observed for tracer particles in active gels [START_REF] Toyota | Non-Gaussian athermal fluctuations in active gels[END_REF] and living cells [START_REF] Park | Metabolic remodeling of the human red blood cell membrane[END_REF][START_REF] Ben-Isaac | Effective Temperature of Red-Blood-Cell Membrane Fluctuations[END_REF][START_REF] Trepat | Physical forces during collective cell migration[END_REF][START_REF] Bursac | Cytoskeletal remodelling and slow dynamics in the living cell[END_REF][START_REF] Fodor | Activity-driven fluctuations in living cells[END_REF]. They are more pronounced in the Rho inhibitor case, as a signature of larger fluctuations, possibly due to directed motion events.

Phenomenological model of vertex dynamics: transient confinements and large displacements

To quantitatively discriminate between the effects of the different inhibitors, we analyze our measurements with a nonequilibrium model previously introduced to describe tracer fluctuations inside living cells [START_REF] Fodor | Activity-driven fluctuations in living cells[END_REF]. This model is not aimed at describing any specific process that produces the active fluctuations, it rather formulates a general framework that allows one to quantify non-equilibrium forces and fluctuations. We regard the vertex as a virtual particle which dynamics is prescribed by two coupled equations: (i) an equilibrium diffusion of the vertex in a cage, modelled as an harmonic potential of stiffness k-the displacement is driven by a Gaussian white noise of variance 2γk B T with a drag force of coefficient γ; (ii) a non-Gaussian colored diffusion equation for the center of the cage, mimicking nonequilibrium activity as a run-and-tumble dynamics. Inspired by the large ballistic-like displacements that we observe in experimental trajectories, we model this active noise as a two-state Poisson process: the cage has a constant velocity v in a random uniformly sampled two-dimensional direction during a random persistence time of average τ , and it remains fixed during a random quiescence time of mean τ 0 . We understand the confinement as an elastic mechanical stress resulting from cells surrounding each vertex, and the nonequilibrium motion of the cage as an active stress. The effect of this active stress is to reorganize the structure of the monolayer, and therefore to spatially redistribute the elastic mechanical stress.

In the absence of activity, this model predicts a short time diffusion, and then a large time plateau expressing the elastic confinement. Such dynamics is entirely under the control of equilibrium thermal fluctuations. In an active system, nonequilibrium processes enhance vertex displacement via the cage motion, yielding a free diffusion of the vertex with coefficient D A = (vτ ) 2 /[2(τ + τ 0 )]. The large time dynamics is fully determined by the active parameters {v, τ, τ 0 }, whereas thermal fluctuations control the short times via {k, γ, T }. A sub-diffusive transient regime appears between the two diffusions, as a crossover towards a plateau, and a super-diffusive regime can also precede the large time diffusion, as a signature of the ballistic motion involved in the active noise. In such a case, thermal effects are negligible at times larger than τ c = τ k B T /(kD A ), a timescale quantifying the transition from the short time equilibriumlike dynamics to the large time active diffusion. Simulated trajectories exhibit clusters of similar size accounting for the transient confinement of the vertex. Occasionally large displacements of order vτ appear. The vertices do not only fluctuate around a local equilibrium position, they also undergo rapid directed jumps (compare Fig. 1(e) and Fig. 3(a)).

In contrast to previous works which describe the manybody dynamics of cells in the tissue in a more complete framework (37-39), we do not explicitly account for interactions between neighbouring vertices. As a result of such interactions, the cells experience an intermittent dynamics alternating between fluctuations of small amplitude and rapid large displacements. As in glassy systems, the jumps appear through collective rearranging regions [START_REF] Bi | Motility-Driven Glass and Jamming Transitions in Biological Tissues[END_REF], thus contributing to the non-Gaussian fluctuations experienced by the vertices. Even though non-Gaussian fluctuations are present independently of topological transitions, the existence of a quantitative connexion between collective rearrangements and such transitions is still an open question [START_REF] Bi | Energy barriers and cell migration in densely packed tissues[END_REF]. Our approach consists in reducing the dynamics of a large number of interacting vertices into the dynamics of a single vertex embedded in an effective background that describes the mean-field effect of the surrounding system. Within our model, interactions are embodied by both the elastic confinement and the active source of fluctuations leading to large displacements. We argue below that our phenomenological approach is sufficient to capture the vertex dynamics, since it provides a framework to decipher the effects of the Rho pathway inhibitions on this dynamics.

Order relation in the active diffusion coefficient and the persistence time

We fit the MSD data with our analytic prediction to estimate the parameters characterizing nonequilibrium activity. Our fits convincingly capture the transition from sub-diffusive to super-diffusive like behaviors [Fig. 2(a)]. Within our model, these behaviors correspond to cross-over regimes between the short and large time diffusion. We extract from the best fits a single set of passive and active parameters for each condition. Our estimate for τ c can be compared with timescales quantifying the transition from elastic to fluidlike behavior of the material (see Tab. S1 in the Supporting Material), which is of the same order as the Maxwell time reported in three-dimensional cell agregates, i.e. about 30-40 min [START_REF] Marmottant | The role of fluctuations and stress on the effective viscosity of cell aggregates[END_REF][START_REF] Forgacs | Viscoelastic Properties of Living Embryonic Tissues: a Quantitative Study[END_REF][START_REF] Guevorkian | Aspiration of Biological Viscoelastic Drops[END_REF][START_REF] Stirbat | Fine Tuning of Tissues' Viscosity and Surface Tension through Contractility Suggests a New Role for α-Catenin[END_REF]. We report clear quantitative variations of both the active diffusion coefficient D A and the persistence time τ for all conditions [Figs. 3(b,c)]. This suggests that our model, based on separating purely active fluctuations from equilibrium thermal ones, is a reliable framework to capture the effects of our inhibitors on tissue dynamics. Note that passive parameters, such as the relaxation time scale τ r = γ/k reported in Tab. S1 (see the Supporting Material), have also different values between the conditions. This reflects the effects of inhibitors on tissue mechanics, showing that inhibitors also affects the characteristics of passive fluctuations. In this respect, D A and τ are the parameters that characterize only the active contribution to vertex fluctuations, which is the main focus of our study.

Strikingly, D A and τ are larger for Rho inhibitor than for Rho kinase inhibitor, and than for direct myosin inhibition [Figs. 3(b,c)]. The more upstream the inhibition along the pathway, the larger the amplitude of fluctuations and the more persistent the ensuing displacement. The myosin inhibition leads to the smallest D A and τ values, suggesting that the mesoscopic activity of vertices is strongly affected. The Rho kinase target directly activates the myosin, but it also inhibits the myosin light chain phosphatase (MLCP) which in turn inhibits the myosin. Therefore, the result of Rho kinase inhibition on myosin can not be anticipated a priori. Our analysis shows that activity of vertices is less affected than for the myosin inhibitor case: the corresponding value of D A for Rho kinase inhibitor is close to the one for control, which suggests a compensation between activation and de-activation of myosin.

Order relation confirmed by persistence lengths

To gain further insight into the active component of the dynamics, we compare the displacement PDF extracted from the simulated trajectories of vertex dynamics with experimental distributions. The distribution at short time is Gaussian and entirely controlled by the passive parameters: the simulations with or without active fluctuations, where we use passive parameters estimated from fits of MSD data, give the same results at short times (see yellow curves in Figs. 2(be)). Including the active component for the dynamics leaves us with one remaining free parameter: the average persistence length vτ of large displacements. The short time Gaussian remains unchanged, whereas exponential tails develop at large times in the simulated PDF. The tails are more pronounced as time increases, while the central Gaussian part barely changes. We adjust the vτ value by matching the tails appearing in numerical results and experimental data.

The simulated PDFs compare very well with experiments at large times, showing that our simulations reproduce the evolution of experimental distributions at all times [Figs. 2(b-e)]. This is one of the main success of our analysis: the phenomenological model on which relies the quantification of internal activity is able to capture the strong non-Gaussian tails of the distribution with only one free parameter. This supports the underlying picture that vertex dynamics essentially alternates between transient confinement and directed motions. The order of magnitude of the extracted mean persistence length vτ is consistent with our measurements [Fig. 3(d)]. We report again the same order relation within the Rho pathway, i.e. an increase of vτ from myosin inhibitor to Rho kinase inhibitor, and from Rho kinase inhibitor to Rho inhibitor, as a signature of enhanced directed motion.

Dissipation is constant for all inhibitors

A major asset of our model is that it allows us to predict the mean rate of energy dissipated by the vertex dynamics in its surrounding environment. It is defined as the difference between the power injected by the fluctuating thermal force and the one that the moving vertex dissipates via the drag force: J = ẋ(γ ẋ -√ 2γk B T ξ) , where ẋ is the vertex velocity, and ξ is a zero-mean Gaussian white noise [START_REF] Sekimoto | Complementarity Relation for Irreversible Process Derived from Stochastic Energetics[END_REF][START_REF] Sekimoto | Kinetic Characterization of Heat Bath and the Energetics of Thermal Ratchet Models[END_REF]. It vanishes for systems in a thermodynamic equilibrium state. The active nonequilibrium fluctuations lead to a non zero dissipation rate: J = kD A /(1+τ /τ r ) [START_REF] Fodor | Energetics of active fluctuations in living cells[END_REF]. This rate of dissipated energy reflects the excess power injected by nonequilibrium internal activity driving the large displacements of vertices.

When computing the dissipation rate in the four conditions, it appears as approximately constant [Fig. 3(e)], in contrast with the order relation found for active parameters [Figs. 3(b-d)]. This supports that the same amount of energy is dissipated by the vertex large displacements, though the features of such displacements differ between conditions. Given that J depends both on parameters of active fluctuations {D A , τ } and on parameters of passive mechanics {γ, k}, our result suggests that there may be an underlying coupling between mechanical properties of the tissue and its nonequilibrium fluctuations. A possible interpretation is that the nonequilibrium processes at the origin of active fluctuations, such as forces induced by myosin and by actin polymerization, might also affect the tissue mechanics in such a way that the dissipation rate remains unchanged over all the conditions. In that respect, we observe that the relaxation time τ r is increased for Rho inhibitor case with respect to others (see Tab. S1 in the Supporting Material), as also observed for the persistence time τ [Fig. 3(c)].

DISCUSSION

The parameters of vertex fluctuations reveal a correlation between the Rho pathway hierarchy and the junction active fluctuations: the higher up the Rho pathway is the inhibition, the lower is the effect on decreasing the fluctuations of the vertices. Such a relation highlights the usefulness of our methodology to probe quantitatively how signaling pathways control emergent fluctuations. In that respect, our approach bridges biochemical signaling pathways with mechanical fluctuations in vivo. It could be used to characterize quantitatively the effect on fluctuations of other inhibitions acting on different signaling pathways.

Our analysis is based on a phenomenological model which deliberately avoids a detailed description of the many processes operating at the subcellular scale. It is also distinct from other models which consider explicit interactions between neighbouring cells. In that respect, our results do not rely neither on the microscopic details of activityinduced forces nor on the form of interactions within the tissue. We rather postulate an effective vertex dynamics by explicitly distinguishing thermal and active fluctuations. As a result, any underlying mechanism which could rationalize the relation between pathway inhibition and mechanical fluctuations is out of the scope of our approach. Yet, the relation between biochemical signaling and active fluctuations that we quantified in tissues could help future studies to understand the precise relation between the subcellular cytoskeleton dynamics and the emergent vertex fluctuations. In that respect, a possible extension of the model could consist in considering more complex activation/inhibition in the Rho pathway and their connection with contractile forces in the cell [START_REF] Besser | Coupling biochemistry and mechanics in cell adhesion: a model for inhomogeneous stress fiber contraction[END_REF][START_REF] Bement | Activatorinhibitor coupling between Rho signalling and actin assembly makes the cell cortex an excitable medium[END_REF].

In order to test the robustness of our model, one could measure the response of vertices to an external perturbation [START_REF] Harris | Characterizing the mechanics of cultured cell monolayers[END_REF]. The intracellular mechanics is viscoelastic in a large variety of living systems. Our phenomenological model has already been extended to account for a complex rheology [START_REF] Fodor | Nonequilibrium dissipation in living oocytes[END_REF]. It would be interesting to determine whether the relation between fluctuations and pathway inhibitions is similar when including viscoelastic effects. Moreover, applying an external potential to confine a vertex, one could extract work from the vertex fluctuations by varying in time the potential parameters. Our framework allows one to predict the details of the extracted work as a function of the active fluctuation characteristics [START_REF] Fodor | Energetics of active fluctuations in living cells[END_REF]. Confronting such predictions with experimental results would provide another test for the validity of our approach.

The response in living systems is not related to the spontaneous fluctuations, in contrast to equilibrium where such a relation is given by the fluctuation-dissipation theorem (FDT). Therefore, by comparing the response with spontaneous fluctuations of tissues, one could quantify the departure from the FDT, namely the deviation of the dynamics from equilibrium. Moreover, recent progress in nonequilibrium statistical mechanics have shed light on the direct relation between the rate of energy dissipated by the internal nonequilibrium processes and the violation of the FDT [START_REF] Harada | Equality Connecting Energy Dissipation with a Violation of the Fluctuation-Response Relation[END_REF]. This has already led to access to the dissipation rate in some biological contexts [START_REF] Fodor | Nonequilibrium dissipation in living oocytes[END_REF][START_REF] Toyabe | Nonequilibrium Energetics of a Single F1-ATPase Molecule[END_REF]. In that respect, comparing response and fluctuations will allow one to propose an alternative quantification of dissipation rates in epithelial tissues.

The Rho pathway is conserved across species, suggesting that regulations of activity in epithelial monolayers may share common pathways in a large variety of tissues and organisms [START_REF] Etienne-Manneville | Rho GTPases in cell biology[END_REF]. Our approach, based on a quantitative analysis of vertex fluctuations, could serve as a novel framework to decipher the complex regulation of spatial fluctuations by the Rho pathway in other model systems. In that respect, it could be used to analyze vertex fluctuations in developing embryos, such as in Drosophila or in C. elegans, where internal reorganizations are driven by spontaneous topological transitions [START_REF] Rauzi | Nature and anisotropy of cortical forces orienting Drosophila tissue morphogenesis[END_REF][START_REF] Guillot | Mechanics of Epithelial Tissue Homeostasis and Morphogenesis[END_REF].

Figure 1 :

 1 Figure 1: Study of vertex fluctuations. (a) Actin (red) and myosin (green) structures at the apical surface of a MDCK cell (scale bar 3 µm). The myosin is concentrated in dense contractile units (yellow arrow) referred to as myosin clusters. (b) Actin structure alone. (c) Myosin structure alone. (d) We visualise MDCK cell monolayer by GFP E-cadherin (scale bar 30 µm). (Inset) We identify the meeting points between three cells as the privileged point for our analysis (scale bar 4 µm). (e) Extraction of a typical transition between two locally stable positions in vertex trajectory (total time 8 hours). (f) Simplified diagram of the Rho pathway installing an order relation in myosin activation, as presented in (16); in red the specific inhibitors and their targets. (g) Area of individual cells in each condition. C: Control; My: Myosin inhibitor; Rk: Rho kinase (ROCK) inhibitor; R: Rho inhibitor. Number of experiments × number of cells = C: 4×235; My: 4×246; Rk: 3×249; R: 3×148. (h) Area of myosin clusters. Number of experiments × number of clusters = C: 2 × 312; My: 2 × 186; Rk: 2 × 197; R: 2 × 257. Statistical analysis with one-way ANOVA test: ns (non significant) p > 0.05, * p < 0.05, ** p < 0.01, *** p < 0.001 (see the Supporting Material).

Figure 2 :

 2 Figure 2: Statistics of vertex displacement. (a) Mean square displacement as a function of time in four conditions: control (black), myosin inhibitor (blue), Rho kinase inhibitor (orange), and Rho inhibitor (red). The corresponding best fitting curves are in solid lines. The blue and red dashed lines report the large time behaviors. (b-e) Distribution of displacement for the four conditions at three times: 5 (•), 25 (+), and 60 min (•). Exponential tails appear at long times as a consequence of directed motion events in vertex dynamics. Results of simulated dynamics are in solid lines.

Figure 3 :

 3 Figure 3: Active parameters of vertex fluctuations. (a) Typical trajectory obtained from simulations of the vertex dynamics in control condition (scale bar 1 µm). Isotropic "blobs" reveal equilibrium-like transient confinement during a typical time τ c (dashed blue box), and large displacements of order vτ occur due to nonequilibrium activity (orange arrows). (b) Best fit values of the active diffusion coefficient, (c) the persistence time, (d) the persistence length, and (e) the energy dissipation rate. Statistical analysis with t-test: ns (non significant) p > 0.05, * p < 0.05, ** p < 0.01, *** p < 0.001 (see the Supporting Material).

Biophysical Journal 00(00) 1-8

Acknowledgements

We thank F. Graner for helpful discussions. We acknowledge J. W. Nelson for sending the MDCK cell lines. We also thank N. Maggartou for extraction of data and lab. members for discussions. This work is supported by CNRS, FRC and University of Strasbourg. N.S.G. is the incumbent of the Lee and William Abramowitz Professorial Chair of Biophysics.

SUPPORTING MATERIAL

An online supplement to this article can be found by visiting BJ Online at http://www.biophysj.org.

Author contributions

V.M., J.C., R.T., and D.R. designed and performed the experiments. É.F., N.S.G., P.V., and F.v.W. designed the model. All authors contributed to analyzing the data and writing the manuscript.