A L Godeau

M Leoni

J Comelles

H Delanoë-Ayari

A Ott

S Harlepp

P Sens

D Riveline
email: riveline@unistra.fr

A scallop theorem for cells moving in 3D

The famous scallop theorem proposed by Purcell in 1977 states that self-propelled objects swimming at low Reynolds number must follow a cycle of shape changes that breaks temporal symmetry. This should hold true for crawling cells as well. However a clear mechanism for this symmetry breaking is still elusive. Here we show that cells embedded in 3D matrix form at both sides of the nucleus force dipoles driven by myosin that locally and periodically pinch the matrix. Using a combination of 3D live cell imaging, traction force microscopy and a minimal model with multipolar expansion, we show that the existence of a phase shift between the two dipoles involves mainly the microtubular network and is required for directed cell motion. We confirm this mechanism by triggering local dipolar contractions with a laser, which leads to directed motion. Our study reveals that the cell controls its motility by synchronising dipolar forces distributed at front and back. This result opens new strategies to externally control cell motion.

At cellular scales inertia is negligible. As a result low Reynolds number micro-swimmers and micro-crawlers must obey the scallop theorem, stating that the sequence of shape changes performed by self-propelled objects cannot be symmetrical in time [1][2][3][4][5][6]. Cells moving in 2D or in micro-channels often display clear spatial polarisation, characterised by F-actin flowing from the front to the back of the cell [7,8]. The scallop theorem is obviously satisfied, and the cell velocity is directly related to the dynamics of the actin flow of the order of µm/min. Many cells, especially mesenchymal cells moving in 3D matrices, do not show such polarisation, but instead display sequences of protrusion/retraction, leading to a much slower net velocity of the order of µm/hr. The mechanism by which time-reversal symmetry is broken under these conditions remains unknown [9]. Basic principles governing 3D motion of mesenchymal cells still remain elusive.

To address this issue we designed a Cell Derived Matrix (CDM) based on [10] (Fig. 1 and Material and Methods I B 1). For migration experiments, cells were embedded within a 13µm CDM layer (Fig. S1). Key molecular players are fluorescently labelled: (i) the fibronectin network within the CDM (Fig. 1a); (ii) components of the cell cytoskeleton, acto-myosin and microtubules (Fig. 1 a,c,d and Movies 3,5); (iii) focal contacts. Focal contacts promote adhesion and must exert local traction forces on the fibronectin network for the cell to move (Fig. S2 and Movie 4).

The CDM behaves as a soft and elastic material of elastic modulus ≈ 50 Pa as determined by beads embedded within the CDM and moved by optical tweezers, see Material and Methods I B 5 and Movie 1. The CDM was easily deformed by cells as they moved (Fig. 1 a,b and . This enabled us to quantify the associated matrix deformation [11] via the Kanade-Lucas-Tomasi (KLT) tracker method [START_REF] Lucas | International Joint Conference on Artificial Intelligence[END_REF] (Fig. 1 b,d). In our assay both the network deformation and the cellular machinery responsible for force-generation could be tracked simultaneously.

During observation some wild-type cells showed persistent motion while others did not move. Interestingly both motile and non-motile cells exhibited local zones of contraction-extension, equivalent to two force dipoles on either side of the nucleus (Fig. 1 b). These deformations were cyclic, with periods of ≈ 10 mins and amplitudes of typically 2.5 µm, reminiscent of previous reports of cell autonomous contractile oscillations [START_REF] Kruse | Current Topics in Developmental[END_REF]. Periods of oscillations were comparable for motile and non-motile cells (Fig. 2 g,i), see Material and Methods I B 6 for quantification of the oscillations. The phases of contraction correlated with the formation of local myosin clusters (Fig. 1 c). This is consistent with the observation of distinct myosin-driven contraction centres in the migration of neurons on 2D surfaces [14]. We observed more protrusive activity at one of the two ends of elongated cells, in both motile and non-motile cells. This apparent spatial polarisation was not sufficient to elicit directed motion. We plotted the temporal cross-correlation function of contraction-extension at the two force dipoles. Strikingly, while non-migrating cells showed no clear phase shift between the two ends, with a cross-correlation peak at t = 0, migrating cells showed a persistent time lag of ≈ 2 min (Fig. 2 i,j). We propose that this lag is a signature of the "scallop theorem" [1], that is, a manifestation of time reversal asymmetry. As shown by Purcell, this requires at least two degrees of freedom. These are provided by the two distinct sub-cellular contractionextension dipoles identified in migrating cells -and the forces driving them.

The most intuitive way to visualise time reversal symmetry breaking is through the existence of cycles in a properly chosen phase space, as was done in the seminal paper of Purcell [1]. This phase space may be based on a multipole expansion of the traction force exerted by the cell on its surrounding. This approach was pioneered by Tanimoto & Sano for Dictyostelium discoideum crawling on deformable 2D substrate [15]. In particular, they showed that cell translocation was correlated with cycles of finite area in the dipole-quadrupole phase space. In the present work, the traction forces on the matrix are analysed indirectly through a multipole expansion of the rate of matrix deformation. Briefly (see details in Material and Methods I B 7), we extracted the substrate ve- with

D ij = n ∆ (n) i u (n) j , where ∆ (n) i
is the i-th component of the vector joining the cell centre and the point n on the mesh. Similarly, the quadrupole tensor is defined as

Q ijk = n ∆ (n) i ∆ (n) j u (n) k
. We then consider the projection of these quantities on the main axis identified by the average direction of the cell trajectory and we call these components the main dipole, D, and the main quadrupole, Q.

We determine the time variation of the dominant components of the dipole and quadrupole matrices. The dipole/quadrupole observed for non-migrating cells are comparable in magnitude to those of migrating cells (Fig. 3 a,b). This shows that the absence of migration is not due to the lack of traction forces (see also Fig. S4).

The cell trajectory represented in the dipole/quadrupole phase space showed a cycle enclosing a finite area for migrating cells, but a vanishing area for non-migrating cells (Fig. 3 c,d, and more examples Fig. S5). To further test the relationship between finite cycle area and motility of individual cells, we analyzed the migration of the same cell that underwent motion and then stopped (Fig. 3 e-i and Movie 7). Strikingly, we observed that the cycles switched from a finite area while the cell was moving to a vanishing area when the cell stopped (Fig. 3 h,i). The finite area is a direct illustration that the phase shift between the contraction-extension cycles at the two ends of migrating cells observed in Fig. 2j,k also manifests itself in the traction force pattern exerted by the cell on the matrix through focal contacts. This is a clear signature of time reversal symmetry breaking.

To determine whether we could relate the correlation between the oscillations at the two cell ends with molecular actors, we tracked cell motion in the presence of the microtubule depolymerizing agent, nocodazole. Nocodazole treated cells undergo oscillations driven by local force dipoles with typical periods ∼ 10min (Fig. 2 g) but do not maintain a fixed cell-polarization (Movie 12). In this case, oscillations at different cell ends were anticorrelated, and cells did not show directed motion. This further supports the notion that a coupling between os-cillators is needed to promote directed motion. This also suggests that the microtubule network, itself, is a crucial element impacting on the correlation and time delay between oscillations across the cell, which in turn break the cell symmetry and promote cell locomotion. This result is consistent with the notion that the microtubular network regulates the polarity of migrating cells [16,17].

The geometry and dynamics of the traction force distribution suggest a direct comparison with models of self-propelled objects made of discrete moving beads [2,3,5,6]. Fig. 4a,b displays an idealised cell with two pairs of beads exerting time-shifted oscillatory force dipoles at its two ends. The cell activity is characterised by the amplitude d and period T of the oscillations and a phase shift ψ = 2π∆T /T between oscillations at the two ends. The simplest self-propelled object is a microswimmer embedded in a fluid matrix and migrating due to hydrodynamic interactions [2,3,5]. In this case, the period of oscillation is the only time scale in the problem and, in the limit of small oscillation amplitudes, the net cell velocity over a cycle follows the scaling [5]

V swim = d 2 L s T f s (ψ) (1)
where f s (ψ) is a periodic function of the phase shift satisfying f s (ψ = 0) = 0, i.e. no velocity without phase shift, as required by the scallop theorem, and L s ∝ r 4 /(aD 2 ) is a length scale entirely set by the cell geometry (r is the extension/contraction rates of dipole "A" and "B" are shown in red and black, respectively, in unit d/T . The cell velocity, calculated using the model discussed in [5], is shown in green in the same units. It oscillates between positive and negative values -with a non-vanishing mean -with a period equal to that of individual dipoles. c) Typical plot of the experimentally measured instantaneous speed of a migrating cell over time, showing oscillation with a non-vanishing mean. d) Persistent speed as a function of speed period for control cells and cells treated with specific cytoskeleton inhibitors: 10µM ROCK inhibitor Y-27632; 10µM MLCK inhibitor ML-7; 100 µM lamellipodia growth promoter C8 [18] and 50 µM Arp2/3 inhibitor CK666, see Material and Methods I B 2. Each data point corresponds to one cell. The plot displays a decay consistent with a power law. The continuous lines show the fits for V ∼ 1/T (dark blue) and V ∼ 1/T 2 (light blue), following Eqs. (1,2).

the distance between the dipoles, D the dipole size and a the bead size -Fig. 4

a).

A key aspect of cell crawling, which is absent for microswimmers, is the dynamics of cell attachment and detachment from the surrounding matrix. Our results suggest that dipole contraction is associated with an active contraction of acto-myosin clusters and that dipole extension corresponds to the elastic relaxation of the CDM following local cell detachment, i.e. the loss of focal contacts, see Movie 4 and [19]. The kinetics of cell binding/unbinding to the extracellular matrix defines additional dynamic parameters, which can at the simplest level, be captured by a velocity scale v adh [6]. For cells crawling on a rigid substrate -and in the limit of small oscillation amplitudes and fast binding kinetics -the net crawling velocity can be written as [6]:

V crawl = d 3 v adh L c T 2 f c (ψ) (2)
where f c (ψ = 0) = 0 as for swimmers, and the lengthscale L c ∝ r 2 /a also includes additional dimensionless factors related to substrate dissipations (see [6] for more details). These simple models predict how the velocity should depend on the period of oscillation T . The result depends on whether the amplitude of oscillations d is fixed or depends on T . For oscillations of constant amplitude, the net velocity of both swimmers and crawlers decreases if the period of oscillation increases. On the contrary, the velocity is expected to increase with the period if the amplitude increases linearly with the period: d ∼ T , as can be expected if the self-propelled object operates under constant force -or equivalently constant contraction/extension rate. The temporal oscillations of the instantaneous cell velocity is a good readout for the dynamics of the internal force generation. For such self-propelled objects, the instantaneous velocity oscillates around the average values given by Eqs.1 or 2 with a time dependence that reflects the dynamics of the underlying force dipoles. An example of such (theoretical) velocity oscillations can be seen on Fig. 4 b. Experimental observations indeed report strong oscillations of the instantaneous cell velocity (Fig. 4 c).

To test how the period of oscillation of migrating cells influenced their velocity, we tracked cell motion for a variety of specific inhibitors that impact on oscillation periods, see Material and MethodsI B 6, Fig. S3 and Movies 10-12. We plotted the cell velocity as a function of oscillation period for all conditions (Fig. 4 d). An inverse correlation between period and velocity is evident from the data. This is consistent with locomotion being driven by controlling cell deformation instead of cell traction forces. Similar conclusions have been reached in the different context of adherent epithelial cells [20].

Altogether, these results suggest that the temporal coupling between spatially distributed force dipoles along the cell promotes cell motion. To verify this, we needed to find a way to externally trigger localised cellular force dipoles. We leveraged the fact that localized laser ablation led to the recruitment of actin cytoskeleton and localised cell contraction associated to a pinching of the matrix (Fig. 5 a-b and Movie 8). We then used this method to locally impose cellular force dipoles, by triggering local contractions alternatively at either sides of the cell. We imposed correlated contractions by selecting a constant time interval between consecutive laser ablations: this triggered translocation. This shows that externally induced force-dipoles are sufficient to promote directed cell motion (Fig. 5 c-d andMovie 9).

Acto-myosin complexes are likely to be the functional elements that control the dynamics of the individual contractile units. Remarkably, the dynamics of individual dipoles appear similar in migrating and non-migrating cells, suggesting that the same force generation machinery is equally active in both types of cells, and that it is the synchronisation between individual units that makes movement possible. Altering the dynamics of individual units can affect motion, in particular, faster oscillations can lead to faster motion, but the coordination between units is key in enabling cell translocation.

Acto-myosin networks commonly show oscillatory dynamics in vitro and in vivo in a variety of systems and over a large range of length scales [START_REF] Kruse | Current Topics in Developmental[END_REF]: single filaments in motility assays [21][22][23], cells [24] and cell fragments [25], and even entire organisms [26]. The biological function of these generic dynamics is often unclear. However in our case, they appear to be essential. Fast oscillations could lead to quicker motions, and we conjecture that cellular systems could adapt their velocity by modulating the oscillation period. In this context, it would be interesting to test this proposal by combining tracking of single moving cells and local matrix deformations in vivo. If confirmed, this would provide an outstanding example of a physiological relevance for such oscillations.

The phase-shift between contractile units encodes cell polarity. Its maintenance in the course of time requires the existence of a polarity memory. If this type of phase locking can be expected in non-linear systems [START_REF] Pikovsky | Synchronization: A Universal Concept in Nonlinear Science[END_REF], it is more demanding in the cellular context, where it is challenged by strong fluctuations in protein concentrations and activities. Our results suggest that the microtubular network is involved. However, other cytokeletal elements and their interplay with adhesion dynamics are likely to play a role as well. Disentangling the interplay between mechanics and biochemical regulation in this process remains an important open question.

We propose that temporal correlations between distinct contraction-extension units along the cell body is a general principle used by mesenchymal cells to achieve directional motility in 3D. This suggests new strategies to control the motion of cell by externally modulating the local contractile activity, for which we give a proofof-principle using a standard laser setup. This concept could also be used to design synthetic micro-crawlers. Whereas there exists many examples of artificial microswimmers (see [START_REF] Elgeti | [END_REF] for a review), there is to our knowledge no realisation of micro-crawler in regimes where inertia is negligible. To perform the measurements at the location where cells apply forces, a 1µm size bead is placed in the CDM and trapped in the laser beam of an optical tweezer, scale bar: 5µm. The stage is moved back and forth (see also Movie 1). Distribution of the Young Modulus shows a mean value of 50 Pa Trapping and untrapping curves of the bead (blue curve) show that bead movement is symmetric, suggesting that the CDM is essentially an elastic material. Bottom row : the CDM keeps its shape over time; CDM was visualized before at t = 0 (red) and after the cell passage at t = 24h (green); the merged image (last panel) shows that the CDM network remains unchanged. Scale bar: 10µm.

B. Material and Methods

Preparation of CDM

For the CDM preparation (see Suppl. Fig. 1a and [19]), a glass coverslip (CS) was incubated with 1% gelatin (gelatin from cold water fish skin, Sigma) and put at 37 o C in the incubator for 1h. After two washing steps with 1% PBS, the CS was incubated with the gelatin solution for 20 min at room temperature with 1% glutaraldehyde (Sigma). The CS was rinsed again twice with PBS before incubation for 20 -30 min with 1 M glycine (Sigma). Subsequently the CS was washed twice with PBS before plating of the NIH3T3 fibroblasts. Cells were plated at high density in order to produce CDMs. For NIH3T3, this corresponded to a cell density of 10 5 cells/mm in the Petri dish. The culture medium was supplemented with 50µg/ml L-ascorbic acid and changed every two days. The culture was maintained for 8 -9 days. Cells were removed by a lysis medium consisting of 20mM NH4OH and 0.5% Triton (both Sigma) in 1x PBS after two washing steps with PBS. The pre-warmed lysis medium was carefully pipetted on the CS and incubated for up to 10 min at 37 o C in the incubator. PBS solution was added and the CDM stored at 4 o C. The day after, the PBS solution was carefully changed three times to remove residues of Triton. The matrices were covered with PBS and stored for up to one month at 4 o C. For alignment purposes after acquisition, beads (200nm, BioSpheres) were spin-coated on the CS before incubation with gelatin. Beads for optical tweezers measurements (L4530, Sigma) were inserted when seeding the cells. For visualization of fibronectin inside the CDM, two methods were used with no apparent differences : we prepared a stable cell line expressing fluorescent fibronectin (construct kindly provided by Erickson laboratory, Duke University), alternatively, fluorescently labelled FN (Cytoskeleton Inc.) was added to the culture.

Cell culture, transfection and drugs

Cells were cultured at 37 o C under 5% CO2 with a culture medium, high glucose D-MEM with 1% penstrep (Penicillin Streptomycin, Fisher Scientific) supplemented with 10% Bovine Calf Serum (BCS, Sigma). Transfections were performed with Lipofectamin 2000 (Invitrogen) using a standard protocol, and the following constructs were used : mCherry Lifeact, GFP-NMHC2A (kindly provided by Ewa Paluch lab, UCL), RFP zyxin (kindly provided by Anna Huttenlocher lab, University of Wisconsin-Madison) or mCherry MRLC2A (Addgene). For experiments with inhibitors, ROCK inhibitor Y-27632 was used at a concentration of 10µM, microtubule de-polymerising agent nocodazole at 10µM, myosin II inhibitor blebbistatin at 25µM, MLCK inhibitor ML-7 at 10µM, F-actin depolymerizing agent latrunculin A at 1µM, Arp2/3 inhibitor CK666 at 50µM (all Sigma) and lamellipodia growth promoter C8 at 100µM [18]. Before drug addition, we performed a control acquisition of at least one hour. To prevent flows, defocusing or potential damage of the CDM during manipulation, medium with drugs was added to the running experiment without removing the medium to reach the target concentration.

Time-lapse imaging and laser ablation, Optical set-ups, Electron Microscopy

We used a Nikon Ti Eclipse inverted microscope equipped with a Lambda DG-4 (Shutter Instruments Company), a charge coupled device (CCD) camera CoolSNAP HQ2 (Photometrics), a temperature control system (Life Imaging Services) and, if needed, CO2 control. The objectives were the following : PhLL 20x (air, 0.95 NA, phase contrast, Nikon), a Plan Apo 60x objective (oil, 1.40 NA, DIC, Nikon) and a x40 (air, 0.95 NA, Olympus) objectives with a home-made adapter to fit the Nikon microscope. Images were acquired with the NIS Elements software (v3.10, SP3, Nikon) and then exported for further processing. We also used a CKX41 inverted phase-contrast microscope (Olympus) with a cooled CCD camera (Hamamatsu). The system was equipped with a temperature control (Cube box system) and 4x, 10x, 20x and 40x phase contrast air objectives. For confocal imaging, we used a Leica TCS SP5-MP inverted microscope equipped with a Leica Application Suite Advanced Fluorescence LAS AF 2.6.3.8173/LAS AF 3.1.2.8785 acquisition system with hybrid detectors (HyD), photomultiplier tube (PMT) and a heating system (Cube box system); with a HCX PL APO 63X oil objective (1.4 NA, Leica). Additionally, a Leica DMI6000 inverted microscope was used as a spinning disk, equipped with a Andor iQ 1.9.1 acquisition system, Yokogawa CSU22 spinning disk unit and a heating system (Tokai Hit Stage Top Incubator, INU incubation system for microscopes). A HCX PL APO 40x oil objective (1.25 NA, Leica) was used. The time between two frames ranged from 10s to 5min and typical exposure time was 100 -200 ms. The software Imaris was used for reconstructing and animating 3D data sets.

For laser ablation, a Leica TCS SP8-MP based on a Leica DM6000 CFS upright microscope was used, equipped with a Leica Application Suite Advanced Fluorescence LAS AF 2.6.3.8173/LAS AF 3.1.2.8785 acquisition system with photomultiplier tube (PMT) and hybrid detectors (HyD) and an environmental chamber for temperature control (Cube box system). A 25X HCX IRAPO L water objective (0.95 NA, Leica) was used and ablation performed with infrared pulsed laser (Coherent Vision II with pre-compensation). FRAP module was used with point ablation with a wavelength set at 800 nm and an exposition time of 100-200 ms. During experiments, parameters such as laser power and gain were set to minimum to have the smallest cut possible while maintaining cell and CDM integrity.

Native CDMs with integrated cells were imaged by electron microscopy (EM). To have the z-information of CDM, they were grown on Polystyrene sheets. The CDMs were fixed by immersion in 2.5% glutaraldehyde and 2.5% paraformaldehyde in cacodylate buffer (0.1 M, pH 7.4), and post-fixed in 1% osmium tetroxide in 0.1M cacodylate buffer for 1 hour at 4 o C. The samples were dehydrated through graded alcohol (50, 70, 90, 100%), and critical point dried with hexamethyldisilazane. After mounting on stubs with conductive carbon adhesive tabs, the CDMs were coated with gold-palladium in sputter coater (BAL-TEC SCD 005). For imaging along the z-axis, CDMs were cut and mounted upright with conductive carbon tape. Then they were examined by XL SIRION 200 FEG SEM (FEI company, Eindhoven, The Netherlands).

Image Analysis

Images were pre-treated for image analysis. In the case of a z-stack, images were projected with average or maximal projection and outliers were removed with Remove Outlier function of ImageJ. Then deformation was tracked in 2D. A "pyramidal implementation" of KLT tracker method was used to detect deformation in the mesh [19]. This method is based on Kanade-Lucas-Tomasi algorithm and follows bright features from one image to another. Therefore, a textured patch with high intensity variation in x and y is required. A multi-resolution pyramid of the image intensity and its gradients are computed before tracking is performed. Then the KLT algorithm is first applied to lower resolution image, where it detects coarse movement before a higher resolution image is taken for fine movement detection. After having reached the maximum iteration steps for all pyramid levels, the displacement of the feature is extracted (between two frames). The Computer Vision Toolbox for MatLab was used with a customary written code with number of features varying between 5000-10000 and a minimum distance from 8-14 px depending on image size, making sure that features were sufficiently spaced. Parameter window interrogation size was set to 40 px and maximal iteration to 20 px. The number of pyramids was two for all calculations. For each image an overlay of displacement vectors and phase contrast or fluorescent image of cell was generated. Drift calculation was performed with maximum of 40 px features with a minimum distance of 8 px with a window size of 20 px and one pyramid. The displacement due to drift was subtracted from the cell induced displacement of the mesh.

Once the displacement of the meshwork calculated via the KLT feature tracker, the displacement of the matrix was projected onto a line going through the cell axis in order to observe 1D matrix displacement and the heatmap of displacement plotted (Fig. 2). The matrix displacement amplitude is colour coded in the heatmap. To highlight the cell position in the heatmap, cell features such as nuclear front, back and cell tail were tracked and plotted in the heatmap. To investigate recurring patterns of matrix displacement along the cell axis, autocorrelation analysis was performed on the projected matrix displacement. The displacement was averaged over a region of 2-5 um either at the cell back or the cell front both regions delimited by nucleus back over a time period where a cell was either migrating or not migrating. Average matrix displacement was subtracted to reduce background and autocorrelation performed via Matlab using the xcorr function with coefficient normalization so that the autocorrelation at zero lag equals 1.The highest peak next to 0 was extracted. Graphs where no peak could be identified were discarded. To compare matrix displacement at two distinct positions along cell axis, crosscorrelation analysis was performed with the same parameters as for the autocorrelation. In this case the highest (or deepest peak) was extracted. Graphs where no peak could be identified were discarded.

Elasticity CDM measurements by optical tweezers

CDM with fluorescent beads were mounted on a holder and placed on an inverted microscope (Olympus IX71). A Spectra Physics YAG laser (1064 nm) was used and focused through a high numerical oil immerse on objective (Zeiss achromat x100 1.25 NA). We acquired the movies with a second objective (Olympus X40 0.6 NA) associated with a CCD camera (DCC3240C, Thorlabs). The setup was controlled by LabView 9 (National Instruments). Beads were centered in the optical trap (Suppl. Fig. 1e). The position of the CS (bottom of CDM) was registered to obtain the z position of the measured beads in the CDM. Stage was moved in 0.2µm/s or 0.1 nm/s steps, covering a distance of 2 -4µm in x/-x and y/-y directions (see Movie 1). The laser power was calibrated with beads in solution. Subsequent data processing was performed with ImageJ bead Tracker Plugin and further post-processed with IgorPro Wavemetrics. ) that minimize the monopole term. We consider regions enclosed by different radii R k , with k = 1, 2, ... and compute the multipoles in each of these regions. We then compute the average values of the multipolar terms and obtain error bars as the standard deviations over all these regions. Insets: average traction, computed as the average of the absolute value of the rate of deformation field within the circular region. The traction in the non-migrating phase is slightly smaller than the traction during the migrating phase but the two values are comparable. (Right): the cell does not move because the topology of the vector field has changed compared to the migrating phase, showing now mainly a dipolar field.

Average values of period, deformation and persistent speed

We quantified the fibronectin network dynamics, Fig. 2c, and we measured periods of 10.2 ± 1.4 mins for the front and 9.9 ± 1.1 mins for the back of the cells and amplitudes of typically 2.5 µm were reached. Non-motile cells also exhibit local spots of contraction-extension, Fig. 3a, with periods of matrix displacement of 8.1 ± 1.3 min for the front and 7.1 ± 1.2 min for the back dipole. For nococazole, cells switched polarity in the course of the experiments and only a single period of 9.5 ± 1.2 min was extracted. In Figure 3c, Persistent speed are: 0.54 ± 0.04µm/min (control); 0.47 ± 0.02µm/min (cells treated with Y-27632), 0.28 ± 0.02µm/min (blebbistatin); 0.43 ± 0.03µm/min (ML-7); 0.39 ± 0.06µm/min (C8) and 0.40 ± 0.03µm/min (CK666). Average oscillation periods are: 15.00 ± 1.25 min (Control); 20.7 ± 1.3 min (Y-27632); 24 ± 3 min (ML-7); 27 ± 3 min (C8); and 27 ± 4 min (CK666).

Multipolar expansion

To quantify the cell-matrix interaction, deformation, we determine the substrate velocity vector field u (n) i as described in the main text. The component i at position n of the mesh of the substrate velocity u (n) i has the same topology of the traction force vector field [15] (which we do not compute here). We then calculate the monopole vector

M i = n u (n) j , the dipole matrix S ij = Dij +Dji 2
and the quadrupole matric

Q ijk = n ∆ (n) i ∆ (n) j u (n) k , where D ij = n ∆ (n) i u (n) j , where ∆ (n) i
is the i-th component of the vector joining the cell centre (defined below) and the point n on the mesh. We also compute to total traction defined as n |u (n) |. We then consider the projection of these quantities on the main axis identified by the average direction of the cell trajectory and we call these components the main monopole, M, the main dipole, D, and the main quadrupole, Q. For cells which are non-migrating (i.e. that do not move at all) but are still elongated we identify the main axis as the cell axis.

A cell that migrates spontaneously (i.e. in absence of external forces) can be viewed as a force-free body, hence in the absence of inertia the sum of all the traction forces add up to zero [15]. Thus the monopole should be zero. In our experiments, 3D imaging is challenged by a limited and asymmetric field of view which lead to a nonzero value of the monopole. To reduce these imaging asymmetries, the traction, monopole, dipole and quadrupole are computed and We observe a variability in the traction forces for both migrating and non-migrating cells, but we exclude that non-migration is due to lack of traction (compare top left and right panels). For non-migrating cells exerting a single force dipole, the quadrupole is expected to be zero. The non-zero quadrupole, seen to be correlated with the dipole for non-moving cells, may be due to asymmetries in the field of view, see above (Section I B 7).

averaged over discs (Fig. S4) of increasing radius R k , k = 0, 1, ...M starting from a minimum radius R 0 ∼ 20µm and up to a maximum radius R M defined as the biggest radius such that the circular region is fully contained within the boundaries of the experimental images. To minimise the spurious monopole, the center of the disc is varied around the apparent cell center (obtained from cell tracking) and the location for which the monopole is minimum is adopted as the center of the discs on which the dipoles and quadrupoles are computed. This is done at every time step.

Additional examples of cycles for migrating and non-migrating cells are shown in Fig. S5. As in the main text, the cycles shown for migrating cells are obtained by first identifying intervals of time with clear oscillating behavior in the speed and focusing on that time interval when we compute the multipolar expansion. For cells which are non-migrating, we cannot apply this scheme as we have no notion of oscillating speed, thus the choice in Fig. S5 is somehow arbitrary. Nonetheless, we find that the areas of cycles for non-migrating cells are systematically negligible when compared to the areas of migrating cells.

C. List of Movies

• Movie 1 : CDM is elastic, as shown by optical tweezer characterization, time in mm:ss.

• Movie 2 : Cell deforms the CDM, time in hh:mm.

• Movie 3 : Cells deform the fibronectin network in yellow while moving, scale bar 20 µm, time in hh:mm.

• Movie 4 : Cells motion in 3D with CDM and the associated focal contacts dynamics (fibronectin in yellow and zyxin in red), time in hh:mm:ss.

• Movie 5 : Microtubule asymmetric distribution (left) is associated to cell polarity during motion, time in hh:mm.

• Movie 6 : The volume of the nucleus impedes cell motion, scale bar 25 µm, time in hh:mm.

• Movie 7 : Phase-shift between local dipoles is associated to cell motion; cell motion (left, LifeAct red), fibronectin network deformation (center yellow), merge, time in hh:mm:ss.

• Movie 8 : Local laser ablation triggers the local recruitment of actin and local contraction, time in mm:ss.

• Movie 9 : Induced dipoles (indicated by arrows) trigger cell motion, time in mm:ss.

• Movie 10 : Cell motility in the presence of blebbistatin added at time 0, time in hh:mm, scale bar 25 µm.

• Movie 11 : Cell motility in the presence of latrunculin A added at time 0, time in hh:mm, scale bar 25 µm.

• Movie 12 : Cell motility in the presence of nocodazole added at time 0, time in hh:mm, scale bar 25 µm.

FIG. 1 .FIG. 2 .

 12 FIG. 1. Key players in cell motility. a) Left panel (and Movie 4): A cell deforms the fibronectin (FN) network when migrating (FN in yellow and mCherry LifeAct for actin filaments in red). Right panel: Enlargement of the white windows of the left panel. Black arrows highlight displacement of fibers due to cell movement. b) Overlay of phase contrast image and KLT calculation of mesh displacement (green arrows -with scale bar shown) with local contraction and extension dipoles indicated with white windows. c) Myosin clusters form locally within cells and are correlated with local contraction. Bottom: KLT deformation (green arrows) and myosin mCherry signal. Top: Enlargement of the white window of the bottom panel showing the myosin mCherry signal. d) α-tubulin staining of a cell inside CDM. Microtubules extend from the centrosome to the periphery of the cell (see also Movie 5). In panels, scale bars: (a) 25 µm (b,c,d) 10 µm.

  locity vector field u (n) i for the component i of the change of substrate deformation between two consecutive frames at position n of the mesh. The dipole is a tensor defined as S ij = Dij +Dji 2

FIG. 3 .

 3 FIG. 3. Multipole analysis of the matrix deformation rate. a) Snapshots of a cell with: matrix rate of deformation, green arrows, the main dipole axis, blue, the axis of the cell motion, red. b) Time series of the main dipole, blue, and quadrupole, magenta, -projected on the cell axis -for a migrating cell (squares) and a non-migrating cell (circles), sampling approximatively 1/10 of the duration of the entire experiment. c-d) Cell trajectory in the dipole/quadrupole phase space for a migrating cell (c) and a non-migrating cell (d). The migrating cell follows a cycle with a finite area and the non-migrating cell does not. The error bars are obtained following the procedure described in Material and Methods I B 7. The individual cycles for different radii are shown in light gray. e) Snapshots of a cell which in the course of the same experiment displays a transition from migrating to non-migrating behaviour (Lifeact in red and fibronectin in yellow), scale bar 10µm. (f-g) Cell positions in the x-y plane (blue and orange curves) during the migrating phase (f -migration mainly along the x-axis) and the non-migrating phase (g -absence of net cell motion). (h-i) Cycles for this cell in the dipole/quadrupole phase space during the migrating (h) and non-migrating (i) phases, showing the same difference in area (see also Movie 7).

FIG. 4 .

 4 FIG.4. Persistent speed is related to the period of oscillations. a) Schematics of dipoles distribution highlighting quantities used in the theoretical model: two dipolar units ("A" and "B") made up of disks of radius a, through which cells exert traction forces on the extracellular environment. The dipoles, at distance r apart, oscillate with period T , with minimum amplitude D and a maximum amplitude D + d. b) Model dynamics. Left: Alternate phases of extension/contraction are imposed to the two dipoles, defining a cycle ("1, 2, 3, 4, 1...") that is not time-reversible. Right: the extension/contraction rates of dipole "A" and "B" are shown in red and black, respectively, in unit d/T . The cell velocity, calculated using the model discussed in[5], is shown in green in the same units. It oscillates between positive and negative values -with a non-vanishing mean -with a period equal to that of individual dipoles. c) Typical plot of the experimentally measured instantaneous speed of a migrating cell over time, showing oscillation with a non-vanishing mean. d) Persistent speed as a function of speed period for control cells and cells treated with specific cytoskeleton inhibitors: 10µM ROCK inhibitor Y-27632; 10µM MLCK inhibitor ML-7; 100 µM lamellipodia growth promoter C8[18] and 50 µM Arp2/3 inhibitor CK666, see Material and Methods I B 2. Each data point corresponds to one cell. The plot displays a decay consistent with a power law. The continuous lines show the fits for V ∼ 1/T (dark blue) and V ∼ 1/T 2 (light blue), following Eqs.(1,2).

FIG. 5 .

 5 FIG. 5. Cell motion is triggered by means of laser induced force dipoles. a) Schematic of laser ablation experiment. b) (Left panel: Lifeact, middle panel: phase contrast and KLT) Ablation at the back of the cell (white arrow) immediately followed by an extension, and later by a contraction of the matrix, (both highlighted using white square window). scale bar LifeAct: 10 µm KLT: 20 µm. Right panel. Bottom: Plot of the displacement rate along the cell axis at different time (colour coded) showing extension and contraction. Top: Heatmap of displacement rates indicating the initial extension and the subsequent contraction. c) Top: sequence of snapshots during laser ablation on cell expressing mCherry LifeAct. Intensity drops locally immediately after the cut, followed by a local recruitment of actin, scale bar 20 µm, scale bar in zoom 5 µm. Bottom: the intensity heatmap reveals a focused actin flow (see also Movie 8). As shown in the deformation map, the contraction precedes this flow. d) Consecutive ablations (indicated with white arrows) mimics contraction-extension cycles at the front and back of cell. Ablation is performed in the following order: at the cell back, at the front and then at the back again. In all panels, scale bar 10µm and time in mm : ss. Note cell motion to the right (see also Movie 9), scale bar 10 µm. The cell is outlined in red and the back of the nucleus with a blue dashed line.

  FIG. S1. Characterization of CDM. a) Schematics of the CDM preparation: Cells are cultured on a gelatin based surface during eight days and then removed by lysis. b) Electron microscope images of the top layer (x -y) and the side view (x -z) of CDM; despite the high density of the top layer, cells enter within minutes; scale bars 10µm. c) Cells are embedded in the CDM. Top: schematic; middle: a typical image with actin in red and fibronectin in yellow; bottom : cell thickness is 7.8 ± 0.3µm compared to CDM thickness 12.8 ± 0.7µm d) Typical image of fibronectin labeling within CDM. Top, view in the x -y plane (scale bar: 20µm) and, bottom, side view in the x -z plane (scale bar: 10µm). e) Top row : Rheological characterization of the CDM by optical tweezer (see Movie 1).To perform the measurements at the location where cells apply forces, a 1µm size bead is placed in the CDM and trapped in the laser beam of an optical tweezer, scale bar: 5µm. The stage is moved back and forth (see alsoMovie 1). Distribution of the Young Modulus shows a mean value of 50 Pa Trapping and untrapping curves of the bead (blue curve) show that bead movement is symmetric, suggesting that the CDM is essentially an elastic material. Bottom row : the CDM keeps its shape over time; CDM was visualized before at t = 0 (red) and after the cell passage at t = 24h (green); the merged image (last panel) shows that the CDM network remains unchanged. Scale bar: 10µm.

  FIG. S2. Cells deform the CDM network by passing through the network and by exerting local forces through focal contacts. FN in yellow and cell expressing RFP-zyxin, scale bar 25µm. Right side shows a blow up of one the zones highlighted by the squares, scale bar 5µm. White arrows show the movement of a FN fiber, taken at two consecutive instants of time. Bottom panels : The enlargement of a pore behind the cell suggests a local pulling force by the cell, mediated by focal contacts. Time in min.

  FIG. S3. Cell motion in CDM is modified in the presence of specific inhibitors. Typical trajectories of the control experiment over 5 hours, in (a), are compared to data acquired in similar conditions but where cell morphologies are altered in the presence of: (b) blebbistatin; (c) Y-27632; (d) ML-7; (e) latrunculin A and (h) nocodazole. Treatments with C8, in (f),[18] and with CK666, in (g), do not alter the cell morphology. Trajectories are shorter for all drugs except Y-27632. No migration is observed in the case of nocodazole and latrunculin A; see also Movies 10-12. Typical cell morphologies are shown for each condition, using bright field, on the top panel, and Life-Act labeling, on the bottom panel. i) Persistent speed for control and cells treated with 10µM Y-27632, 25µM blebbistatin, 10 µM ML-7, 100 µM C8 and 50 µM CK666. (Average values are reported in Material and MethodsI B 6 j) Average of the period of the cell speed calculated by autocorrelation analysis of speed projected on axis of migration. Scale bars 25µm.

  FIG.S4. Method to compute the multipolar terms illustrated by using the same cell of Fig.4(b), which in the course of the same experiment shows a transition between migrating phase, (left), and non-migrating phase, (right). (Left): experimental image showing a cell in light blue, and the vector field restricted to a circular region used to compute the multipoles. The center (red cross) is obtained starting from the cell centre position and searching for the coordinates (X * , Y * ) that minimize the monopole term. We consider regions enclosed by different radii R k , with k = 1, 2, ... and compute the multipoles in each of these regions. We then compute the average values of the multipolar terms and obtain error bars as the standard deviations over all these regions. Insets: average traction, computed as the average of the absolute value of the rate of deformation field within the circular region. The traction in the non-migrating phase is slightly smaller than the traction during the migrating phase but the two values are comparable. (Right): the cell does not move because the topology of the vector field has changed compared to the migrating phase, showing now mainly a dipolar field.

  FIG. S5. Comparison between cycle curves obtained in different experimentsfor different migrating (left panels) and nonmigrating (right panels) cells. We observe a variability in the traction forces for both migrating and non-migrating cells, but we exclude that non-migration is due to lack of traction (compare top left and right panels). For non-migrating cells exerting a single force dipole, the quadrupole is expected to be zero. The non-zero quadrupole, seen to be correlated with the dipole for non-moving cells, may be due to asymmetries in the field of view, see above (Section I B 7).

ACKNOWLEDGMENTS

We thank the Riveline Lab. for discussions and help, M. Maaloum, the Imaging and Microscopy Platform of IGBMC, and H.P. Erickson, A. Huttenlocher, E. Paluch for constructs, J. Goetz for the former CDM protocol and M. Piel for critical reading of the manuscript. D.R. acknowledges support from CNRS (ATIP), ciFRC Strasbourg, the University of Strasbourg, Labex IGBMC, Foundation Cino del Duca, Région Alsace, Saarland University. A. Ott and D. Riveline acknowledge support from DFH-UFA through the Collège Doctoral Franco-Allemand CDFA-01-13. This study with the reference ANR-10-LABX-0030-INRT has been supported by a French state fund through the Agence Nationale de la Recherche under the frame programme Investissements d'Avenir labelled ANR-10-IDEX-0002-02. M.L. acknowledges financial support from the ICAM Branch Contributions and Labex CelTisPhyBio N o ANR-10-LBX-0038 part of the IDEX PSL N o ANR-10-IDEX-0001-02 PSL. A. Ott acknowledges support by DFG within the collaborative research center SFB 1027.


Amélie Luise Godeau

Hélène Delanoë-Ayari

Daniel Riveline

Amélie Luise Godeau

Generation of fluorescent cell-derived-matrix to study 3D cell migration

Keywords: biological physics, cell motility, traction force microscopy, cell derived matrix, fibronectin, cytoskeleton

Cell migration is involved in key phenomena in biology, ranging from development to cancer. Fibroblasts move between organs in 3D polymeric networks. So far, motile cells were mainly tracked in vitro on Petri dishes or on coverslips, i.e. 2D flat surfaces, which made the extrapolation to 3D physiological environments difficult. We therefore prepared 3D Cell Derived Matrix (CDM) with specific characteristics with the goal of extracting the main readouts required to measure and characterise cell motion: cell specific matrix deformation through the tracking of fluorescent fibronectin within CDM, focal contacts as the cell anchor and acto-myosin cytoskeleton which applies cellular forces. We report our method for generating this assay of physiological-like gel with relevant readouts together with its potential impact in explaining cell motility in vivo.

Introduction 1.Cell migration : 2D versus 3D

Cell migration has been characterised on Petri dishes and 2D flat surfaces for more than a century [START_REF] Grote | Petri dish versus Winogradsky column: a longue durée perspective on purity and diversity in microbiology, 1880s-1980s[END_REF][START_REF] Petri | Eine kleine Modification des Koch'schen Plattenverfahrens[END_REF]. In this condition, cells are easily tracked within single focal planes with phase contrast microscopy. Their culture is accessible to any laboratory with standard equipment and since the seminal studies of Michael Abercrombie [START_REF] Abercrombie | Observations on the social behaviour of cells in tissue culture: I. Speed of movement of chick heart fibroblasts in relation to their mutual contacts[END_REF], this test remains the classical and 'easy' assay for the study of cell motility.

However, cells evolve in 3D environments in vivo, i.e. usually in dense polymeric networks. This change in geometry and dimensions implies numerous differences between motions in 2D and in 3D conditions : (i) distributions of adhesion zones -focal contacts -are located at the cell-substrate interface in 2D and all around the cell surface in 3D; (ii) compositions of adhesion contacts can be distinct; (iii) 3D networks entail physical constraints for cells [START_REF] Charras | Physical influences of the extracellular environment on cell migration[END_REF] which need to generate their own space by potentially squeezing nuclei [START_REF] Wolf | Physical limits of cell migration: control by ECM space and nuclear deformation and tuning by proteolysis and traction force[END_REF] or by cleaving their environments with metalloproteinase [START_REF] Nagase | Structure and function of matrix metalloproteinases and TIMPs[END_REF]; (iv) 2D flat surfaces are very rigid compared to soft physiological matrices, and this can change motion and the associated dynamics of the cytoskeleton.

These differences come along with distinct cellular structures responsible for cell motility. In the classical view on 2D surfaces, cells adhere, they generate lamellipodia at their front which extend through actin polymerisation, myosin II based contraction contributes to deform the cell body forward, and focal adhesions at the back of the cell are disassembled [START_REF] Ridley | Cell migration: integrating signals from front to back[END_REF]. For 3D environments, the picture is more complex and cells show multiple strategies to move. Indeed, cell migration in 3D has been studied in different configurations: in gels [START_REF] Lämmermann | Rapid leukocyte migration by integrin-independent flowing and squeezing[END_REF][START_REF] Petrie | Nonpolarized signaling reveals two distinct modes of 3D cell migration[END_REF][START_REF] Wolf | Physical limits of cell migration: control by ECM space and nuclear deformation and tuning by proteolysis and traction force[END_REF], in micro-channels [START_REF] Lautenschlager | The regulatory role of cell mechanics for migration of differentiating myeloid cells[END_REF][START_REF] Liu | Confinement and Low Adhesion Induce Fast Amoeboid Migration of Slow Mesenchymal Cells[END_REF][START_REF] Prentice-Mott | Biased migration of confined neutrophil-like cells in asymmetric hydraulic environments[END_REF], or cells 'sandwiched' between two plates [START_REF] Yip | Traction stress analysis and modeling reveal that amoeboid migration in confined spaces is accompanied by expansive forces and requires the structural integrity of the membrane-cortex interactions[END_REF]. In these assays, various types of migration were reported, such as mesenchymal migration, amoeboid migration [START_REF] Liu | Confinement and Low Adhesion Induce Fast Amoeboid Migration of Slow Mesenchymal Cells[END_REF], or bleb-based [START_REF] Petrie | Nonpolarized signaling reveals two distinct modes of 3D cell migration[END_REF] motion. Each type of migration involves distinct distributions of forces applied by cells (see for example [START_REF] Bergert | Force transmission during adhesion-independent migration[END_REF][START_REF] Liu | Confinement and Low Adhesion Induce Fast Amoeboid Migration of Slow Mesenchymal Cells[END_REF]). This variety of behaviours calls for new assays to connect cellular activity, local force application and cell motion.

During movement, cells apply local forces, which can be decomposed in moments (force monopole, dipole, quadrupole) to generate a motion with zero total force, as set by laws of motion at low Reynolds number [START_REF] Tanimoto | A simple force-motion relation for migrating cells revealed by multipole analysis of traction stress[END_REF]. To identify the composition of these forces, traction force microscopy allows to capture deformations in the local cellular environment. Physical descriptions have emerged for explaining motion in connection with force patterns or cellular structures [START_REF] Saez | Is the Mechanical Activity of Epithelial Cells Controlled by Deformations or Forces?[END_REF][START_REF] Tanimoto | A simple force-motion relation for migrating cells revealed by multipole analysis of traction stress[END_REF].

Along these lines, we decided to design an assay where essential readouts would be acquired and integrated, i.e. matrix deformation and cellular actors generating motility, to understand cell migration. We adapted a CDM protocol [START_REF] Cukierman | Taking cell-matrix adhesions to the third dimension[END_REF] where cells move within a thin layer of physiological matrix in order to visualize the main anchor protein for cells in the extracellular matrix, i.e. fibronectin (FN). We plated cells and observed their motion and 3 the associated deformation of the FN network over hours and days. Altogether, this assay captures spatio-temporal evolution of deformation, anchoring points in the extracellular matrix (ECM), focal contact and acto-myosin distributions. This approach could serve as a generic method to study and characterise cell motion in 3D environments in vitro.

The need for in vitro assays

In order to understand cell motility in 3D, there is a necessity to design specific culture set-ups allowing to control the mechanical and biochemical environments. The mechanical state can be tuned by adjusting Young modulus of the matrix ,and the biochemical composition and state can be modulated by controlling synthesis and growth conditions. To study the implication of forces on cell motility, a substrate or network which can be deformed by cells is required. Considering that cells apply pN forces [START_REF] Balaban | Force and focal adhesion assembly: a close relationship studied using elastic micropatterned substrates[END_REF][START_REF] Riveline | Focal contacts as mechanosensors: externally applied local mechanical force induces growth of focal contacts by an mDia1-dependent and ROCK-independent mechanism[END_REF], the gel should have a low and known Young's Modulus in order to detect forces applied by cells. Cell derived matrices (CDM) have a considerable potential for cell studies in 3D physiological environment [START_REF] Cukierman | Taking cell-matrix adhesions to the third dimension[END_REF]. These matrices are produced by cells, which are removed after days thereby leaving a 3D-like physiological gel. This set-up has already been used for studying focal adhesion distributions in 3D [START_REF] Cukierman | Taking cell-matrix adhesions to the third dimension[END_REF]. Designing a cellular environment with low Young's modulus and a visualisation of the relevant extracellular matrix protein -fibronectin -allows to address specific questions regarding symmetry breaking, spatio-temporal force patterns and cell adhesion in 3D. We achieved this goal by adapting cell derived matrix protocols, which have been shown to generate low Young's modulus with suitable size to embed cells within CDM while allowing migration along x-y plane fitting the cell body.

We prepared and visualised this fibronectin network in two ways. We introduced fluorescently labelled fibronectin to the matrix culture, and we generated a genetically modified cell line which produces and secretes fluorescently labelled fibronectin. We report these methods in this article. In the next section (section 2), we introduce the general matrix protocol, and we explain how to visualise the fluorescent fibronectin within these matrices. This is followed by the description of introduction of beads in the CDM for its mechanical characterisation. In section 3, we explain how to integrate cells in our CDM, and how to set up a time-lapse movie. We also give an example of how to process the data via the Kanade-Lucas-Tomasi (KLT) feature tracker to extract matrix deformation. In section 4, we present and discuss some of our results, and in section 5, we give an outlook of potential applications.

Generation of fluorescent CDM

In the following section, we first explain the method for culturing CDM [START_REF] Cukierman | Taking cell-matrix adhesions to the third dimension[END_REF][START_REF] Castelló-Cros | Extracellular Matrix Protocols[END_REF][START_REF] Goetz | Biomechanical remodeling of the microenvironment by stromal caveolin-1 favors tumor invasion and metastasis[END_REF]. We then describe (i) how to modify this protocol in order to obtain fluorescent matrices; and (ii) how to integrate micron-sized beads within the matrix to generate an external framework for correcting the sample drift or for mechanical characterisation.

Preparation of CDM

Material

Method: Matrix culture

These steps are conducted in sterile conditions under a laminar flow hood. All reagents have been autoclaved or filtered before usage to prevent contamination. In this protocol, we take advantage that fibroblasts produce and secrete extracellular matrix proteins. These proteins are then organized into a physiological-like matrix by cells. To keep and stabilise this matrix, cells are cultured on a gelatin-based support. The protocol is schematically represented in Figure 1 : seeding of cells on the gelatin-based substrate in order to obtain a confluent monolayer, culture of CDM and cells removal. 1.) For microscopy, we use glass coverslips (CS) as a substrate to ease experiments with high resolution. Multiwell plates or Petri dishes can be used as well. A glass coverslip (#1) is covered with 1ml of 1% gelatin (gelatin from cold water fish skin, Sigma) and placed at 37°C in the incubator for 1 h to promote anchorage of the future CDM. After the CS has been washed twice with 1% PBS, the gelatin is incubated for 20 min at room temperature with 1% glutaraldehyde (Sigma). This chemical agent crosslinks gelatin and provides stability to the scaffold. The CS is Seed 3T3 fibroblasts on gelatine based substrate to obtain confluent monolayer

Cells produce and secrete matrix proteins

Remove cells by lysis: CDM rinsed again twice with PBS before incubation for 20-30 min with 1 M glycine (Sigma) at room temperature. This allows to saturate the aldehyde groups generated by the glutaraldehyde treatment in gelatin. Subsequently the CS is washed twice with PBS. At this step, the protocol can be continued, or alternatively the CS can be stored at 4°C up to one month. 2.) A Petri dish with an 80% confluent monolayer is taken and washed 1x with PBS. PBS is removed and trypsin-EDTA solution is added. The dish is incubated for 3 min at 37°C in the incubator to detach cells from the surface. The supernatant is recovered and added to a 15ml tube with DMEM supplemented with 10% BCS and 1% penicillin-streptomycin solution. Cells suspended in cell culture medium are centrifuged for 5 min, the supernatant removed and the cell pellet resuspended in DMEM supplemented with 10% BCS and 1% pen-strep. Cells are plated on the gelatin-coated CS. They need to be plated at high density in order to produce CDM. For NIH3T3 cells, this corresponds to 1 million cells for 35 x 10 mm Petri dish. 3.) This sample is placed in the incubator at 37°C to let cells adhere. Every second day, the culture medium supplemented with 50 μg/ml L-ascorbic acid is changed. L-ascorbic acid is added to promote collagen secretion of cells. The culture is maintained for 8-9 days. During this time, the synthesis and accumulation of CDM can be observed.

Generation of a fluorescent matrix

In the following section, we present two methods to generate cell derived matrices with fluorescently labelled fibronectin. Their quality can be evaluated with immunofluorescence microscopy by comparing the spatial localisations of fibronectin with both labels (data not shown). We add a rhodamine labelled fibronectin to the culture medium (Figure 3a) or we modify genetically cells for making them secrete fluorescent fibronectin (Figure 3b).

Material

Method

1.) For this first method, step 1 and 2 are performed as described in section 2.1.3.

Step 3 is modified as follows: The culture is scaled down to fit a 96 well-plate. Normal NIH3T3 fibroblasts are used for CDM production. Additionally, to the L-Ascorbic acid the medium is supplemented with rhodamine labelled fibronectin (1 μg per 100 μl well) for every change of medium. Cells producing CDM incorporate to the matrix the fluorescent fibronectin present in the medium. After 8-9 days of culture, we proceed to step 4 as described above. When storing at 4°C, the Petri dishes or multiwell plates containing the CDM are surrounded with aluminum foil to prevent photo-bleaching.

2.)

The second method consists in using genetically-modified cells which produce and secrete fluorescently labeled fibronectin. Protocols for establishing stable cell lines can be found elsewhere [START_REF] Ruedel | Transfection Methods[END_REF]. Briefly, we transfected NIH3T3 fibroblasts with a fluorescent Ypet-FN construct from Erickson laboratory [START_REF] Ohashi | Dynamics and elasticity of the fibronectin matrix in living cell culture visualized by fibronectin-green fluorescent protein[END_REF] using Lipofectamin (Invitrogen Gibco, Thermo Fisher Scientific). The method follows the steps in section 2.1.3, but cells stably expressing FN-Ypet are seeded onto the CS for the CDM production in step 3, instead of using regular NIH3T3 fibroblasts. To select cells encoding the Ypet-FN, cell culture media contain G418 antibiotics. After cloning, this stable cell line is used for the preparation of CDM. The fluorescently labelled fibronectin is then secreted and spontaneously incorporated the CDM. When the matrix has been produced, cells are removed as described in step 4 of section 2.1.3. During storage, we surround the Petri dishes or multiwell plates containing the CDM with aluminum foil to prevent photo-bleaching.

Insertion of beads in the CDM

Beads within matrices can be used for different purposes. They can serve as markers for calculating the spatial drift potentially occurring during time-lapse experiments. They can also be used as traps for optical or magnetic tweezers for mechanical characterisation of CDM, or also as fiducial markers to calculate the matrix displacement generated by external forces. We recommend to use monochromatic beads to make sure that there is no colour overlap between fluorescent channels and facilitate further analysis.

Material

Method

1. For drift correction, beads need to stick to the surface of coverslips and not to the matrix, nor to be loosely trapped within the matrix. The coverslips need to be activated by plasma cleaning to make the glass hydrophilic. A bead solution is prepared with a dilution of 1:1000 (to be adjusted with bead size). The CS is placed on a spin-coater and a drop of approximately 200µl of the bead solution is spin-coated at 1000 rpm for 1 min on the CS, and the sample is dried in an oven at 65°C for at least 30 min. Then CS with beads are exposed to UV light for at least 15 min to prevent contamination during CDM culture. Since bead fluorescent intensity is high and stable, the risk of photo-bleaching is low. Afterwards, the protocol from section 2.1.3 can be followed.

2. There is also the option to incorporate fluorescent or magnetic beads in the matrix. Therefore, here again step 3 of the protocol in section 2.1.3 is modified. Fluorescently labelled beads (or any other beads) are added to the culture medium. As described before for fluorescently labeled fibronectin, beads integrate the CDM while cells secrete the matrix proteins. Beads are incorporated throughout the height/thickness of the CDM. It may happen that incorporated beads are not well anchored to the matrix and loosely diffuse within the meshwork. Fluorescent beads can also be used to calculate displacements applied to the matrix and serve as proxys for the matrix/fiber localization. They can as well be employed for mechanical characterisation of the matrix such as magnetic or optical tweezer measurements.

Characterisation of cell migration in fluorescent CDM

When the fluorescent CDM is produced, cells can be incorporated within the matrix. Their motility and force application can then be studied. In the following section, we explain how to integrate cells in the CDM, how to track their migration and the matrix deformation under the microscope, and how to process the data in order to extract the matrix displacement. It is highly recommended to use cells which have been transfected with a construct leading to the expression of relevant fluorescent proteins: we transfected for proteins of the cytoskeleton (actin, microtubule), motor proteins (myosin) or proteins revealing focal adhesions (zyxin, see Figure 5)). This allows to visualise not only the fibronectin of the matrix but also cellular structures which generate forces on the CDM and to follow quantitatively the spatio-temporal behavior of these physical readouts.

Integration of cells in the CDM

Cell medium and trypsin are heated in a water bath to 37°C and CDMs placed at room temperature for at least 1h. The microscope is pre-heated to 37°C for at least 1h (and if possible overnight) to reach stable imaging conditions and to minimise drift during the acquisition.

Material

Method

We prepare cells in a Petri dish reaching 80% confluent monolayer. We then plate cells at 100 cells/mm 2 density. PBS is removed, and the CDM is incubated with cells for at least 2h at 37°C. CS with CDM and cells is mounted on a metallic holder which has to be cleaned with ethanol in a sonicator, dried and then exposed to UV light to reduce possible contamination during experiment. If the microscope is used with CO2, the medium is changed to DMEM supplemented with 1% BCS and 1% penicillin streptomycin; otherwise, we used L-15 medium supplemented with 1% BCS and 1% pen-strep allowing for CO2-free experimental conditions. Subsequently CDM with cells is mounted on the microscope and time-lapse experiment is started. A variety of cells move within these CDMs, stable cell lines, primary cells, fibroblasts and epithelial cells (see Figure 4 and Movies 2-6). For the investigation of cell integration in CDM, cells are resuspended in L15 after trypsinisation and added to the CDM under the microscope. The rapid incorporation of cells within the matrix can be observed as shown in Figure 5 and Movie 7.

Cell migration within CDM

Cell migration is recorded via time-lapse experiments using microscopes. For high resolution and limitation of photo-bleaching, we recommend confocal microscopy and spinning disk microscope. To capture cell migration and deformation of the matrix, a z-stack spanning the CDM thickness is recorded. The frequency of image acquisition is adapted to experimental requirements. At least two channels are used during experiments to differentiate the matrix from cells. In order to have an optimal experiment, photo-bleaching, photo-toxicity and temporal requirements need to be balanced and frequency of image acquisition, exposure time and intensity need to be adapted. As an illustration, typical migration of a cell expressing RFP-zyxin in red in a CDM with fluorescent fibronectin in yellow is shown in Figure 6 and in Movie 8.

Calculation of matrix displacement

To assess the deformation or the displacement of matrix, different methods can be used, such as Particle Imaging Velocimetry (PIV) [START_REF] Pakshir | Dynamic fibroblast contractions attract remote macrophages in fibrillar collagen matrix[END_REF][START_REF] Steinwachs | Threedimensional force microscopy of cells in biopolymer networks[END_REF], other detection methods [START_REF] Legant | Multidimensional traction force microscopy reveals out-of-plane rotational moments about focal adhesions[END_REF][START_REF] Legant | Measurement of mechanical tractions exerted by cells in three-dimensional matrices[END_REF] or custom KLT feature tracker (Godeau et al., in preparation). Before these calculations, images are pre-treated (i) to generate images on which matrix displacements are then evaluated and (ii) to correct for any artifact due to the characteristics of the imaging system or the imaging conditions.

Image pre-treatments

Before extracting the deformation or displacement of the matrix, the raw data of matrix needs to be either pre-treated or processed. The aim of the pre-treatment is to obtain 2D images for the calculation of x-y displacement of the mesh with the best possible signal-to-noise ratio. The pre-treatment options consider images acquired with multi-positions, z-stack and images with artifacts.

• If images have been obtained with multi-positions, stitching of the images needs to be performed to obtain a continuous image. • If a z-stack has been acquired, the different planes need to be projected on one plane. This can be done using Fiji with the z-projection plugin with maximal or average projection. • If artifacts like 'comets' or bright spots can be seen in images, they can be removed by choosing the "remove outlier" or "de-speckle" option in Fiji noise plugin. • If the frame of the image is very large compared to the area of deformation, it can be resized to reduce calculation load of the next step.

Tracker algorithm

In the following, we focus on the KLT tracker method as PIV methods have already been discussed elsewhere [START_REF] Pakshir | Dynamic fibroblast contractions attract remote macrophages in fibrillar collagen matrix[END_REF][START_REF] Steinwachs | Threedimensional force microscopy of cells in biopolymer networks[END_REF]. A "pyramidal implementation" of KLT tracker method can be used to detect deformation in the matrix. This method is based on Kanade-Lucas-Tomasi algorithm developed by Kanadé and Lucas which follows bright features from one image to another [START_REF] Lucas | An Iterative Image Registration Technique with an Application to Stereo Vision[END_REF][START_REF] Tomasi | Detection and Tracking of Point Features Technical Report CMU-CS-91-132[END_REF]. It extracts local regions of interest from each image and identifies the corresponding feature in every image. By comparing two consecutive images or an image compared to a reference image, the displacement can be calculated. The pyramid implementation means that first a multi-resolution pyramid of the image intensity and its gradients are computed, before tracking is performed. Then coarse-grained movement is detected on a lower resolution image, and subsequently higher resolution image is taken where the fine movement is detected. The customary code was written for Matlab.

First, the program identifies either bright features or patches with high intensity variations in x and y. Figure 7a shows the detection of bright features in our matrix. The minimum distance as well as the maximum number of points to be detected can be varied. The number of features to detect depends on image size and quality of the matrix signal. Then, the multi-resolution pyramid of the image intensity and its gradients are computed before tracking is performed. The feature detection is performed on a window on the lower resolution image and then on the higher resolution image. After having reached the maximum iteration for all pyramid levels, the displacement of the feature is extracted (between two frames). An overlay of displacement vectors and phase contrast or fluorescent image of cell can be generated (see Figure 7b). This allows to see if the code runs correctly or if artifacts are generated. The calculation can be corrected for drift by using still beads grafted onto the CS as a reference (see section 2.2) or an area where no cell traction force is applied. The drift is then subtracted from the initial mesh displacement result. From then on, further quantification of the matrix displacement can be performed to identify potential patterns.

Discussion

The combination of fluorescent fibronectin within the CDM with the KLT algorithm provide a high accuracy in measurements of displacements for the study of cell motions in 3D physiological environments. Our results indicate that cells apply a specific spatio-temporal force pattern to break the symmetry and migrate (Godeau et al., in preparation). We observed that cells exhibit a specific oscillatory force pattern with localised contraction centers, where a phase shift between front and back contractions accounts for the motion.

In this study, we used NIH 3T3 fibroblasts for CDM culture and for motility assay. In our CDM preparations with NIH3T3 fibroblast, we obtained elastic matrices. Our CDMs were quite soft ( ̴ 50Pa elastic modulus, Godeau et al, in preparation) in comparison to other CDMs reported in the literature [START_REF] Petrie | Nonpolarized signaling reveals two distinct modes of 3D cell migration[END_REF][START_REF] Soucy | Microelastic properties of lung cellderived extracellular matrix[END_REF][START_REF] Tello | Generating and characterizing the mechanical properties of cell-derived matrices using atomic force microscopy[END_REF]. Furthermore, we did not detect matrix modification due to cell passage. Our assay may be different for cells which were reported to migrate while deteriorating the ECM [START_REF] Wolf | Physical limits of cell migration: control by ECM space and nuclear deformation and tuning by proteolysis and traction force[END_REF]. These mechanical characteristics are expected to depend on the cell type used for CDM synthesis. Each cell type might incorporate different or a different ratios of extracellular matrix proteins and crosslinkers, and these molecular compositions determine the CDM mechanics.

We tested various cell types which all integrated and migrated within the CDM and deformed the matrix (see Fig. 4). The amplitude of forces applied by cells depend on the cell types. However, all cells integrated in the CDM should go through the spatial network constructed by the CDM-generating cells. This also means that the degree of confinement may vary depending on the cell type. Larger cells in CDMs produced by smaller cells may lead to higher confinement, whereas smaller cells in CDMs produced by larger cells may lead to lower confinement.

Conclusion

In this chapter, we presented a protocol to generate fluorescent 3D cell derived matrices which allow to perform traction force microscopy and observe key components for the characterization of motion, fibronectin tracking, focal contacts and acto-myosin cytoskeleton. The protocol as it is described was

Figure 1 :

 1 Figure 1: Schematic description of CDM production. The glass coverslip is covered with gelatin and further treatments with glutaraldehyde and glycine are performed before cells are seeded. Culture is maintained 8-9 days before cells lysis, leaving only the secreted proteins forming the CDM.

  4.) Cells are removed by lysis with a medium consisting of 20 mM NH4OH and 0.5 % Triton (Sigma) in 1x PBS after washing twice with PBS. NH4OH stabilises CDM in the presence of Triton. The prewarmed lysis medium is carefully pipetted on the CS and incubated for up to 10 min at 37°C in the incubator. The cell lysis (Figure2and Movie 1) is rapid, and the extrusion of nuclei can be observed. If the CDM culture (step 3) has been maintained for too long, the CDM might become too dense and nuclei might stay trapped within the CDM. PBS is added and the CDM stored overnight at 4°C : this ensures complete lysis of cellular membranes and provides more temporal stability to the CDM. The day after, PBS is carefully changed 3 times to remove all residues of Triton to avoid undesired effects on cells during experiments. Matrices are covered with PBS and kept at 4°C. Although CDM can be stored for up to 1 month, it is recommended to use them in the following two weeks, as risks of detachment of the matrix from the CS increase with time.

Figure 2 :

 2 Figure 2: Lysis of cells in CDM: images of cells lysis producing CDM. Most of the membranes are lysed within the first 2-3 min of the process and after 10 min no cell membranes or nuclei are visible anymore; see also Movie 1.

Figure 3 :

 3 Figure 3: Fluorescent fibronectin within CDMs. a) Image of fibronectin mesh acquired with a x20 objective. Fluorescent fibronectin added to culture medium is integrated in matrix. b) Image of fibronectin mesh acquired with a x20 objective secreted by genetically modified cells. These cells synthesise and secrete fluorescent fibronectin which is incorporated in the matrix.

oFigure 4 :

 4 Figure 4: Different cell types migrate in CDM: a) NIH3T3, b) HeLa, c) mouse lymphocytes, d) mouse embryonic fibroblasts and e) REF52 fibroblast integrate and show motility in our CDM, see also Movies 2-6. All cell positions are indicated with a red arrow some cells are outlined in light blue.

Figure 5 :

 5 Figure 5: Cells enter the CDM. a) bright field image and b) fibronectin. At time zero, the cell is spherical on top of the CDM; entry occurs within minutes; see also Movie 7.

Figure 6 :

 6 Figure 6: Cell migrating in CDM: a) Time-lapse images of a cell expressing RFP-zyxin a protein of focal adhesions in red, migrating in the fluorescent fibronectin in yellow embedded within the CDM. Time in hh:mm. b) Image sequence of focal adhesion dynamics (top) and growth highlighted in the zoomed region above white arrow for reference, and (bottom) corresponding matrix displacement during cell migration, with enlarged region (right, matrix displacement highlighted with red arrows); see also Movie 8.

Figure 7 :

 7 Figure 7: KLT analysis: a) Example of bright point identification of KLT analysis on FN network of CDM with minimum parameter separation of 20 px. b) Snapshot of cell embedded in matrix imaged with phase contrast (left) and the overlay of the matrix displacement (right).