Supplementary Figure 1. Calibration of the AFM experiments using Formvar sheet device.

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2 Supplementary Figure 1. Calibration o the AFM experiments using Formvar sheet device. a Scheme o the experimental setup o Protrusion Force Microscopy PFM using Formvar sheet-coated grids. Cells were plated on a Formvar -coated grid. The grid was turned upside down, and then ixed on a 4 mm diameter glass coverslip with pieces o double-side tape and immersed in 5 µl o culture medium in the BioCell TM AFM chamber. b AFM measurement o Formvar sheet thicness see Methods, AFM experiments section or details. A representative AFM topographical image o a Formvar sheet layered on a glass coverslip. The mean thicness e o Formvar sheets was calculated by analysis o cross sections 1 measurements per experiment. In this example the thicness was estimated at 61 +/- 7 nm, n=4. c Podosome stiness and height measurements in macrophages plated on the Formvar sheet or on glass serving as a reerence. The device was not reversed as or protrusion measurements. AFM stiness and height measurements were perormed as or macrophages plated on glass coverslips see Methods or details. Means +/-s.e.m o podosome height and stiness are shown. t-test was used to compare at least the stiness o 39 podosomes and the height o 15 podosomes in cells obtained rom at least three donors. P <.5; ns, non-signiicant. d AFM characterization o the compliance o the Formvar sheet. To control that the observed podosome protrusions were not due to a passive compliance o the Formvar sheet, a range o scanning orce rom.1 to 1 nn were used to image ive protrusions on the Formvar sheet in living upper panels and ixed macrophages bottom panels. To control that the orce applied by the AFM cantilever had no irreversible eects on the Formvar sheet, dierent orces were applied sequentially: 5 pn, 1 pn, 1 pn and pn. We veriied that the use o dierent orces applied sequentially did not aect the measurement o protrusion heights o podosomes as shown on the AFM topographical images, indicating that the Formvar sheet is only deormed through active cell processes. Bars: µm.

3 Supplementary Figure : Characterisation o the Formvar elastic properties. a-b Evolution o the F/h ratio applied orce over the membrane delection as a unction o the square o the membrane delection h membrane thicness e =61 nm and grid side a=9 µm. a Case o large delections 5<h<5 nm. Example o three tests perormed onto dierent membranes. According to relation 4 Supplementary Note 1, the slope o the curve allows to calculate the biaxial Young s modulus E/1-. b Case o small delections h<5 nm exerted at the center illed circles or at µm rom the center open circles, allowing the determination o the membrane tensile stress and o the the model thans to relation 4 Supplementary Note 1. C coeicient o

4 Supplementary Figure 3: Characterization o the polyacrylamide hydrogel rigidity. Matrix response to oscillatory shear was measured using a rheometer. Polyacrylamide hydrogels PAs o dierent proportion o acrylamide/bis-acrylamide solution were subjected to oscillatory shear by rotating the cone around its axis with a maximum strain o.5. Shear storage moduli G measured in the linear viscoelastic domain as a unction o the angular velocity rom.1-1 rad.s -1 or the Polyacrylamide gels o dierent compositions used in this study. The Young s modulus indicated in the legend o each curve has been determined as three times the measured value o G see the Methods section.

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6 Supplementary Figure 4: The substrate stiness inluences the ormation, organization and stiness o podosomes. a Representative conocal luorescence images o F-actin red, vinculin green and myosin IIA blue o ixed human macrophages plated on ibrinogen-coated glass coverslips Glass or on Fg-PAs with dierent stiness values 1 Pa, 5 Pa, 6.5 Pa,.5 Pa. Mislocalization o vinculin is observed when the substrate stiness decreases below 1 Pa. Organization o cortical myosin IIA is aected rom 6.5 Pa and below. Arrowheads point to single podosomes. Bars: 1µm, inset bar: µm. b Representative curves o Young s modulus vs indentation depth o individual podosomes monitored in macrophages plated on glass coverslips coated with ibrinogen or Fg-PAs 5 Pa and 6.5 Pa, dashed line shows the average indentation depth above which Young s modulus values reach a plateau.

7 Supplementary Figure 5. Inluence o substrate stiness on podosome stiness a Illustration o podosome stiness measurement by Atomic Force Microscopy AFM. The method used has been described in 1. b-c Podosome stiness b, Young s Modulus E in Pa and height c measured by AFM in living human macrophages plated on glass coverslips coated with ibrinogen or Fg-PAs 5 Pa and 6.5 Pa. Means +/-s.e.m. are shown. Dunnett's multiple comparison test was used to compare experimental data to glass control and t-test was used to compare the variable stiness conditions. The stiness o at least 36 podosomes and the height o at least 133 podosome in cells obtained rom our donors were measured. P <.1; P <.1; ns, non-signiicant. d Representative Young s modulus podosome variations over time in living human macrophages plated on ibrinogen-coated glass coverslips or Fg-PAs 5 Pa and 6.5 Pa. e Quantiication o stiness oscillation periods o podosomes in macrophages plated on dierent substrates coated with ibrinogen. Means +/-s.e.m. are shown. t-test and Dunnett's multiple comparison test were used as described in legend b-c to compare at least 3 podosome oscillation periods measured in cells obtained rom our donors. P <.1; P <.1; P <.5.

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9 Supplementary Figure 6: Podosome ultrastructure. a Scanning electron micrographs o unrooed macrophages highlighting the architecture and interconnections o podosomes through F-actin cables, bar: 1µm. b TIRF images o a macrophage stained or F-actin red, vinculin green and myosin IIA blue, bar: 1 µm. c Upper panels Average o 981 podosomes rom 9 dierent cells and representative o 3 independent experiments made rom 3 dierent donors stained or F-actin, vinculin and myosin IIA and imaged by TIRF microscopy, bar: µm. Lower let panel The luorescence intensity proile extracted rom the dashed-line see upper panels shows that myosin IIA is present outside the vinculin ring. Right panel Scanning electron micrograph o an unrooed macrophage with myosin IIA immuno-gold labelling yellow dots, showing that myosin II is excluded rom podosome cores and accumulates at peripheral actin cables, bar: 1 µm.

10 Supplementary Figure 7: Cytoseleton-disturbing drugs aect the protrusive activity o pododomes. a-b Representative AFM delection images o the Formvar sheet deormed by living macrophages, beore let panels and ater min o CD µm a, bar: 1 µm or ML7+Y763 named ML7+Y, µm +1 µm b, bar: µm.

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12 Supplementary Figure 8: Inluence o cytoseleton-disturbing drugs on human macrophage morphology observed with luorescence microscopy and AFM. a Representative conocal luorescence images o human macrophages plated on glass coverslips, ixed and stained or F-actin blue, myosin IIA red and vinculin green in control condition, or ater treatment ML7+Y763, µm+1 µm. Only a partial mislocalization o myosin IIA can be observed ater ML7+Y763 treatment see enlarged images. Bars: 1µm. b AFM delection images were acquired on living human macrophages plated at least or h on glass coverslips, in control condition or treated with ML7+Y763 or 5 min. Then, macrophages were ixed and stained or F-actin blue, vinculin green, myosin IIA red. Merge reers to the three luorescence staining images and overlay reers to the AFM delection image and the merged luorescence images. For the ML7+Y763 condition, the same cell was imaged beore and ater drug treatment. Bars: 1 µm. c Representative immunoblottings using antibodies directed against PhosphoSer19-MLC P-MLC S19 on cell lysates rom macrophages in control condition or treated with ML7+Y763. d Quantiication o relative PhosphoSer19-MLC levels normalized by actin immunoblot signal used as a loading control. Means +/-s.e.m. are shown. Dunnett's multiple comparison test P <.1 was used to compare immune signal intensity in cell lysates obtained rom three donors.

13 Supplementary Figure 9: Cytoseleton-disturbing drugs aect podosome ormation and stiness. a FRAP analysis o macrophages expressing mrfp-actin and treated with the indicated drugs showed that ML7+Y763 did not signiicantly aect the t 1/ o luorescence recovery o podosomes. Means +/-s.e.m. are shown. Dunnett's multiple comparison test were used to compare the t 1/ o luorescence recovery in at least 37 macrophage podosomes rom at least three donors. P <.1; ns, non-signiicant. b Podosome characterization in living human macrophages. Quantiication o podosome height, stiness and stiness oscillation period in control and drug-treated macrophages. see Means +/-s.e.m. are shown. Dunnett's multiple comparison test were used to compare at least 45 podosome heights, 4 stinesses and 11 stiness oscillation periods rom cells obtained rom at least our donors. P <.1; P <.1.

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15 Supplementary Table 1: Rheology o polyacrylamide hydrogels % Acrylamide % Bis-acrylamide Young s modulus +/- S.D. E in Pa PA 1Pa /- 6 PA 5Pa /- 1 PA 6.5Pa /-.3 PA.5Pa /-.1

16 Supplementary Note 1 Quantiication o the mechanical properties o the Formvar sheet The elastic properties biaxial Young s modulus E/1-υ and average residual stress σ o the Formvar sheet were characterized using the point-delection method on 3+/-3nm-, 61+/-7nm- and 17+/-7nm-thic, 9µm-wide membranes. It consists in determining the stiness o the tested membrane by applying a small concentrated load F at its center by means o an AFM cantilever and in measuring the corresponding out-o-plane displacement h -5.For a clamped membrane the general relationship between the applied load and the maximal displacement is given by 3,4 : 3 3,, 1 1 h a Ee e h C h g a Ee F 1 where a is the membrane s size, e its thicness, E its Young s modulus, υ its Poisson s ratio, α a constant see Supplementary Table and g a unction allowing to tae into account any possible residual in plane stress o the membrane. For an unstressed membrane g =1 while in case o a tensile residual stress σ -4 : E e a Ln K I I K g with K and K 1 are the Bessel unctions o the second ind o orders and 1, I and I 1 the modiied Bessel unctions o the irst ind o order and 1. and are Euler s constant and a coeicient see Supplementary Table, respectively. The C coeicient in relation 1 is written as 4 : / 1 1,, g e h C C e h C 3 The / e h C unction is unnown but stands as a decreasing unction o the h/e ratio. For circular and square membranes the values o the coeicients α, β and C are given in Supplementary Table.

17 Supplementary Table : Values o the, and C coeicients o the model. Shape α β C Circular 1/ Square Taing relation 3 into account, equation 1 becomes: F h Ee C C h / e 3 C h h a g 1 a g 1 a Ee Ee This relation is the sum o three terms which become relevant in three dierent domains o vertical displacement: - For very small delections h/e <.3, the two last terms o equation 4 are negligible and F/h is a constant. I the value o the biaxial Young s modulus is nown then the value o the residual stress can be accurately determined through the unction g 4. - For intermediate delections.3 < h/e < 15, the three terms have to be considered and due to the C h / e unction which is relatively unnown, relation 4 cannot be directly exploited. However, since C h / e is a decreasing unction o h/e and its variations are small over a small domain o displacement h/e <1,, C C is assumed to be h/ e constant. Moreover, on this small domain h/e <1 the last term o relation 4 is negligible, such that the F/h=h representation is linear. The intersection with the Y axis o this straight line enables to determine the value o the residual stress σ through the unction g. - On the other side, or very large delections h/e >15 the second term o equation 4 is negligible and the slope o the representation F/h=h leads to the determination o E/1- υ. The strategy adopted or the determination o both the biaxial mechanical modulus and the residual stress, consists in perorming experiments at large delections 1<h/e <5 to calculate the biaxial modulus rom the slope o the F/h=h representation or h/e >15 and,

18 nowing this modulus, to perorm other experiments at very small delections h/e <1 to estimate the residual stress thans to the intersection with the Y axis o the F/h=h representation. The large delection experiments have been carried out using a PSIA XE 15 AFM in spectrometry mode orce-displacement where the spectrometry curve is a plot o the orce F between the AFM tip and the sample versus the extension o the scanner h. A non-contact mode cantilever 91M-NCHR has been utilized. The exact stiness S c o the cantilever has been determined thans to the F=h curve obtained on a rigid body Silicon. In that case only the cantilever is deormed and it was ound that S c =39+/-.5 N/m. Then, the coupled stiness S o the cantilever in contact with the membrane has experimentally been determined rom the slope o the F= h curve and the stiness S o the membrane 1 was calculated as: S 1 1. S S c Indeed, the cantilever stiness is notably higher than the one o the membranes in a ratio o about 3 and thus S ~S. The estimated error is lower than 5%. Dierent membranes with e =61 nm 13 membranes, 3 nm 4 membranes, and 17 nm 5 membranes have been tested. According to relation 4, Supplementary Fig.a shows the F/h=h representation or three dierent membranes. The slope o these curves or h>1. µm maes it possible to determine the mean value o the Young s modulus and it was ound that E=.1 +/-.51 GPa i υ=.3. This value does not depend on membrane thicness and is in a airly good agreement with those reported in the literature or the bul material.5<e<3 GPa 6. Experiments with very small delections h/e <1 have been carried out using a NanoWizardII TM AFM and, contrary to the previous procedure, the tests have been perormed in the temperature-controlled cell in liquid. Briely, an AFM silicon nitride cantilever with a nominal stiness o.3 N.m -1 with glass beads attached.5 µm diameter was purchased rom Novascan Technologies Inc. Cantilever sensitivity and stiness have been calibrated as mentioned above. In order to estimate the characteristics o the membranes, ten delection curves have been acquired on three points o the Formvar membranes one at the center and two at b=+/- µm rom the center or each applied loading orce: F=.1,.5, 1 and 1.5 nn. The maximal delection h o the probe was then extracted or each curve and a slight dierence ~1 % has been observed as a unction o

19 the location o the measurement Supplementary Fig.b. Indeed, theoretically the ratio o the stiness at the distance b to that at b= is given by: S S b b 1 b 1 a 5 For b=+/- µm this ratio is equal to 1.55 which is in agreement with the experimental value o the order o 1.3. As previously shown and according to relation 4 the F/h=h representation should be linear, which is veriied in Supplementary Fig.b. The membrane stiness S i.e. F/h at h= is equal to nn.nm -1 which gives g = , thus =16 and σ =.1 MPa. The last term o equation 4 is negligible and the slope o the straight line case with b= leads to the C value: C ~7.8 In conclusion, the useul parameters to determine the orces generated by the podosomes onto the Formvar membranes are: the biaxial Young s modulus o the membrane E/1- υ =.3 GPa, the tensile residual stress σ =.1 MPa and the constant C =7.8. Furthermore, rom the experimental value o g, the stiness S, which corresponds to the irst term o equation 4, was determined equal to 6 nn.µm -1, 1 nn.µm -1 and 15 nn.µm -1 or the 3, 61 and 8 nm-thic membranes, respectively.

20 Supplementary Note Estimation o podosome orce To estimate rom the height proiles the orces generated by the podosomes, we considered a simpliied model Fig.5 and section Model or podosome oscillations which describe the interaction o the dierent parts o a podosome F-actin core and F-actin cables with the membrane. With respect to the F-actin core o radius b b~5-3 nm and its close environment, we have perormed detailed calculations assuming either a distributed orce over a dis o radius b or simply a punctual orce b=. The membrane was assumed to be locally clamped by the actin cables. According to the hypothesis o a circular membrane β= and α=1/64π in Eqs1- in the section Quantiication o the mechanical properties o the Formvar sheet and or the smallest displacements h/e <.3, the general shape o the proiles is given by: with,, 4 31 m m E F a h a b a x g h a b a x g Ee F a h 1 where a is the diameter o the protrusion and x the distance rom the center o the protrusion <x<a. The g x/a,b/a unction depends on the boundary conditions o the problem at hand. For a membrane punctually loaded at its center the solution is given by: E e a a x Ln a x K K a x I I I K g with 1 8. The deinition o the dierent Bessel unctions is given in the section Quantiication o the mechanical properties o the Formvar sheet. However, as shown in the section Quantiication o the mechanical properties o the Formvar sheet, the residual stress σ in the membrane is small, consequently is equally small and the shape o the protrusion Eq.1 can by approximated by 7

21 x b x b h hm g, hm g, 3a a a a a This solution corresponds to the proile o an unstressed membrane x/a,b/a loaded in its center multiplied by the g unction. For the unction two cases have been considered Eqs3b-c: Ib : x b b x b x x or x b:, hm 1 1 Ln 3b a a a a a a I b = the membrane is punctually loaded, the previous relation is simpliied: x x x x hm 1 Ln a 3c a a a To tae the small local inclination o the membrane into account, the global equation o the proiles is recast as: x b h cx d, 4a a a where the unctions are given by Eqs3a-c. Fitting the proile o podosome-generated protrusions as measured by AFM with the equations Eqs3a-c and 4a with b=5 nm or b=, we ound that these dierent relations can airly well reproduce the shape o the protrusions. However, the values o h max and a depend on the chosen relation. An example with Eq.4b b= is shown in Fig.1e. x x x 31 a F h cx d hmax 1 Ln with hmax g g 3 a a a 4b 4Ee This procedure has been applied to determine h max and a or all the studied proiles. As is small <1 the g unction.94<g <1 can be airly well approximated by 7 : g 1 with =

22 For the dierent cases, nowing h max, a, g, b b=5 nm or b=, E/1- and e, the orce F can be calculated. With b=5 nm it is shown that as soon as b/a>3 the calculated orces are close to those obtained with b=. As the smallest measured values o b/a are in the range 1.8- Fig. 1g and as generally b/a>3, we have considered a simpliied model with b= case o a punctual orce. Taing relation 5 into account, the general equation 4 small and large delections in the section Quantiication o the mechanical properties o the Formvar sheet is recast as a sum o ive terms: 3 4Ee F 31 a with h h max, C 16 e h C C C Ee h 41 a 58 and 18 h 3 3C h e 3 CEe 41 a h 3 6. The numerical application o equation 6 or 3<e <8 nm, 3<h<7 nm and 45<a<18 nm shows that the two last terms are negligible with respect to the others terms. Hence, the inal equation or the determination o the orces generated by the podosomes is given by: F 4Ee 16 e 3 3 h max hmax h 7 max 31 a C Ee 41 a where E/1-υ =.3 GPa, C=58, =.1 MPa and =18,with e the membrane thicness and with h max and the radius a being measured or calculated rom the proiles o the protrusions Eq.4b.

23 Supplementary Note 3 Model or podosome oscillations The aim o this section is to present a phenomenological model or podosome oscillatory dynamics. The model is based on two main components: a core o polymerizing actin ilaments, whose growth and retraction dynamics depends on the sustained stress, and a set o stress iber-lie structures o actin/myosin that act as tensile or active springs. We show that the coupling o both structures can be generically unstable and generate spontaneous oscillations. The model is ormally close to previous wors on assemblies o molecular motors, which are nown to display spontaneous oscillations beyond a Hop biurcation. One central hypothesis o this model is to consider each podosome as isolated. This is a strong approximation since we neglect here all the possible mechanical interactions between podosomes mediated by the long actin cables connecting the sel-assembled networ o podosomes. However, we will show that despite this approximation, most o the local behavior o podosomes can be resumed by our model. a. Orders o magnitudes Observations presented in the main text show that the typical orce generated by a single podosome varies rom 9 to 155 nn on substrates with stinesses ranging rom 6 to 15 nn.µm -1. Given the contact area o the podosome with the substrate ~.78 μm 8 and the section o a single actin ilament 3.8x1-5 µm 9, this amounts to a orce ranging rom 1.4 to 7.6 pn per actin ilament. This value is close or larger than the stall orce observed experimentally or estimated theoretically 1,11 or single actin ilaments. Interestingly, podosomes have a striing dierence with ocal adhesions. Focal adhesions exert locally a pulling orce on a substrate orce monopole, and orce balance is realized only globally, cell-wide by summing over all FAs. In deep contrast, podosomes exert both a pushing and a pulling orce on the substrate, so that orce balance can be realized at the level o a single podosome orce multipole. This hypothesis o orce balance will be crucial below, and naturally implies that both pushing mechanisms actin polymerization and pulling mechanisms actin/myosin activity are directly coupled and cannot wor independently. b. Model

24 Actin core -- We consider that the actin core is composed o N parallel actin ilaments aligned along the z axis, with their barbed ends pointing towards the substrate in the +z direction. N is irst assumed as constant; its dynamics will be discussed later on. Following the literature, we assume that the dynamics o a single ilament is well captured by a twostate model 1. Each ilament is either in a growing or in a shrining phase, each characterized by a velocity v g and v s respectively. In the growing phase, v g is a decreasing unction o the load applied on the ilament. We assume the orce-velocity relation as given and we will mae use below when needed o a simple linear approximation by writing: 1 where s is the stall orce. We next denote by λ c the rate at which a ilament switches to the shrining phase, which is an increasing unction o the load or >. We will assume here a linear dependence and write: Where H denotes the Heaviside unction. In turn, v s and λ r, which denotes the rate at which a ilament switches rom the shrining phase to the growing phase, are constant since ilaments are not in contact with the substrate in the shrining phase. We irst write the dynamics o the raction nt = Nt/N o ilaments in the growing phase. This reads: 3 We assume that all growing ilaments are in contact with the substrate growing ilaments that are not in contact grow much aster and are assumed to catch up instantaneously with the substrate. The total orce exerted by the growing ilaments along the z direction then reads F =N n, and is assumed to be balanced by the contractile orce exerted by the surrounding contractile actin cables local equilibrium assumption in an inertia-ree medium. Contractile actin cables -- We use standard models o active gel theory to model the actin/myosin structures that surround the actin core. We assume that such structures behave as stress ibers, which have been shown to be well described by classical linear springs 13. We thereore write the projection along z o the total orce in the actin cables as F c = c z c where c is a phenomenological coupling that depends on the myosin activity and taes into account a geometrical actor coming rom the angular distribution o actin cables,

25 which we do not mae explicit here and z c denotes the elongation o the actin cables. Note that c is expected to be proportional to the number o actin cables N c so that we will conveniently write c ~ N when needed. Taing into account the deormation o the c c substrate assumed linear and thereore characterized by an eective spring constant s, which depends on the Young s modulus and tension o the substrate see the section Quantiication o the mechanical properties o the Formvar sheet, and denoting the total elongation by z = z c + z s the orce balance along the z axis inally reads 4 Identiying v g = t z we end up with the ollowing dynamics: [ ] 5 which gives a minimal set o equations modelling the podosome dynamics. Stability analysis: Hop biurcation -- These equations are similar to those obtained in the context o assemblies o rigid molecular motors with load sharing. The ixed point o equation 5 above is given by: [ ] Linearization then yields: [ ] 6 Seeing or a solution or nt and zt proportional to e st yields the ollowing dispersion relation: [ ] [ ] 7 The dynamics is thereore characterized by a Hop biurcation whose critical point is deined by Res=. The ixed point is unstable and spontaneous oscillations occur in the regime [ ] 8 The period o oscillation at the critical point is then given by [ ], 9

26 which can be rewritten as [ ] 1 to mae the dependence on N, N c explicit. c. Discussion The main outcome o this model is the existence o a Hop biurcation with spontaneous oscillations o nt and zt, whose dependence on the model parameters is analyzed below. Note that quantitative predictions will not be discussed here since the parameters λ c, λ r, α, β, s that characterize actin polymerization dynamics are diicult to estimate in in vivo conditions, and c is unnown to the best o our nowledge. The dependence o equations SI 1b 8 and SI 1b 9 on the parameter: 11 shows that the dynamics critically depends on the actin content o the podosome encoded in N, N c and on the stall orce s. Oscillations require that the eective stiness κ be small enough, as can be seen rom Equation 8. This condition is realized i the total actin content is large enough N, N c large, so that κ s /N and oscillations appear. Equation 1 then shows that the period o oscillation grows with N. Non-trivial eects arise rom the act that the actin content is regulated. One can hypothesize, as it is usually done in the context o stress ibers and ocal adhesions, that the size that is, total actin mass here parameterized by N, N c o the podosome depends on the substrate stiness s the larger s the larger N, N c, and that this mechanosensing mechanism is myosin-dependent. This maes the dependence on s o T non-trivial, since s and N s and N c s grow with s. d. Comparison with experiments Stiness and normal orce oscillations -- It is ound experimentally that both the stiness and the normal orce o a podosome oscillate. The observed period is similar although simultaneous observation was not technically possible. The above model predicts spontaneous oscillation o the total deormation z and thereore the podosome height, the

27 F-actin content and the normal orce o the podosome in agreement with these observations. Although the stiness o the podosome does not enter directly the model, it can be argued that stiness increases with the amount o F-actin and internal tensile stress. Additionally, a mechanosensitive recruitment o actin could induce oscillations o the diameter o the actin core coupled to the oscillation o the normal orce. Dependence on actin content: eect o CD treatment -- CD treatment was shown to reduce podosome height, stiness and oscillation period. CD treatment slows down actin polymerization and thereore reduces N as well as v g and thereore s. Under this hypothesis, the model predicts indeed that the normal orce and thereore stiness is reduced. This is in agreement with the height measurements under CD treatment. The above scaling κ s / N obtained in the regime N, N c large shows together with equations 9, 1 that the period o oscillation is decreased when N is decreased by CD treatment, as observed experimentally. Note that under CD treatment oscillations were observed only in the case o height measurements. The model predicts that height and normal orce are directly coupled, and should thereore behave similarly. In this case the amplitude o height oscillations might be too small to be resolved. Note also that the model predicts that CD treatment stabilizes the system by reducing s and N. See condition 8. In this case the ixed point identiied above might be stabilized and no oscillations are expected. Dependence on myosin activity -- It is ound experimentally that inhibition o myosin activation induced a slight decrease o podosome height, stiness, and period o oscillation. The classical hypothesis that the mechanosensing mechanism o stress ibers is myosindependent suggests that the actin content and thereore N and N c depends on myosin activation. Inhibiting myosin should thereore reduce N and N c. Classical models o stress ibers also indicate that c increases with myosin activity. Inhibition o myosin activation thereore clearly reduces the average normal orce as discussed above, and consequently the height and stiness o the podosome. The above scaling κ s /N also shows together with equation 9 that the period o oscillation is decreased when myosin activation is inhibited, as observed experimentally. As stated above we expect that height and normal orce should behave similarly, so that height oscillations should exist under myosin inhibition i podosome height oscillations are observed. As above we note that myosin inhibition stabilizes the system by reducing N, and is thereore liely to abolish oscillations.

28 Eect o substrate stiness -- Observations show that podosome protrusive orce and stiness are increased when substrate stiness is increased. Myosin activity is also increased. As discussed above we invoe a standard mechanosensing mechanism and hypothesize that increasing s increases N and N c. The model is thereore in accordance with our observations as it predicts that the normal orce increases as well as the podosome stiness. Last, we note that the eect o substrate stiness on the period o oscillation would require a reined nowledge o the dependence o N and N c on s, so that its discussion in the ramewor o this model seems too speculative.

29 Supplementary Reerences 1. Labernadie, A., Thibault, C., Vieu, C., Maridonneau-Parini, I. & Charriere, G.M. Dynamics o podosome stiness revealed by atomic orce microscopy. Proc Natl Acad Sci U S A 17, Hong S., W.T.P., Bravman J.C., Nix D. Measuring stiness and residual stresses o silicon nitride ilms,. J. Electr. Mater. 19, Poilane C., D.P., Lexcellent C., Hayashi S., Tobushi H. Analysis o the mechanical behavior o shape memory polymer membranes by nanoindentation, bulging and point membrane delection tests,. Thin Solid Films 379, Martins P., D.P., Malhaire C., Brida S., Barbier D. Bulge test and AFM point delection method, two technics or the mechanical characterisation o very low stiness reestanding ilms. Eur. Phys. J. Appl. Phys. 45, Marutsya S., J.C., Pius Y., Tsuru V.V. Freely suspended layer-by-layer nanomembranes: testing micromechanical properties. Adv. Funct.l Mat. 15, Salih I.K., S.N.J., Abass F.A. Preparation and characterisation o polyvinyl ormal suitable used or wire enamel.. Hadhramout Studies Res.. 7. Timosheno, S.P. & Woinowsy-Krieger, P. Théorie des plaques et coques, Linder, S. Invadosomes at a glance. J Cell Sci 1, Spudich, J.A., Huxley, H.E. & Finch, J.T. Regulation o seletal muscle contraction. II. Structural studies o the interaction o the tropomyosin-troponin complex with actin. J Mol Biol 7, Pesin, C.S., Odell, G.M. & Oster, G.F. Cellular motions and thermal luctuations: the Brownian ratchet. Biophysical Journal 65, Footer, M.J., Kerssemaers, J.W., Theriot, J.A. & Dogterom, M. Direct measurement o orce generation by actin ilament polymerization using an optical trap. Proc Natl Acad Sci U S A 14, Dogterom, M. & Leibler, S. Physical aspects o the growth and regulation o microtubule structures. Phys Rev Lett 7, Trichet, L. et al. Evidence o a large-scale mechanosensing mechanism or cellular adaptation to substrate stiness. Proc Natl Acad Sci U S A 19,