Extracellular matrix in multicellular aggregates acts as a pressure sensor controlling cell proliferation and motility

Imposed deformations play an important role in morphogenesis and tissue homeostasis, both in normal and pathological conditions. To perceive mechanical perturbations of different types and magnitudes, tissues need appropriate detectors, with a compliance that matches the perturbation amplitude. By comparing results of selective osmotic compressions of CT26 mouse cells within multicellular aggregates and global aggregate compressions, we show that global compressions have a strong impact on the aggregates growth and internal cell motility, while selective compressions of same magnitude have almost no effect. Both compressions alter the volume of individual cells in the same way over a shor-timescale, but, by draining the water out of the extracellular matrix, the global one imposes a residual compressive mechanical stress on the cells over a long-timescale, while the selective one does not. We conclude that the extracellular matrix is as a sensor that mechanically regulates cell proliferation and migration in a 3D environment.


Introduction
Aside from biochemical signaling, cellular function and fate also depend on the mechanical state 26 of the surrounding extracellular matrix (ECM) (Humphrey et al., 2014). The ECM is a non-cellular 27 component of tissues providing a scaffold for cellular adhesion and triggering numerous mechan-28 otransduction pathways, involved in morphogenesis and homeostasis (Vogel, 2018). An increasing 29 number of studies in vivo and in vitro shows that changing the mechanical properties of the ECM 30 by re-implanting tissues or changing the stiffness of the adherent substrate is sufficient to reverse  33 The importance of the mechanical context in cancer has been highlighted for a long time by experi-  70 To test the hypothesis that cells respond to 71 the ECM deformation, we introduce an experimen-  88 We developed a simple method to either selectively compress cells embedded in ECM or the whole 89 aggregate composed of ECM and cells. This method is based on the use of osmolytes of different 90 sizes. When big enough, the osmolytes do not infiltrate the ECM and thus compress the whole 91 aggregate by dehydrating the ECM, which in turn mechanically compresses the cells ( spheroids. The effect was visible starting from Π = 500 Pa and saturated at Π ≃ 5 kPa. Unless 97 explicitly stated, the experiments described in this article were performed at Π ≃ 5 kPa, a value 98 that minimizes the pressure, but exacerbates the biological effects. 99 We validated our approach by compressing ECM, cells and multicellular spheroids (MCS) using 100 osmolytes with gyration radii respectively larger and smaller that the ECM pore sizes (figure 2). 101 As osmolytes, we used dextran molecules ranging from 10 to 2000 kDa. As a proxy of ECM,  the mechano-sensitive nature of ion channels. 146 We show in Appendix A.5 that, for realistic estimates of the model parameters, the application 147 of Π with both small or big dextran leads to the same cell volume loss which does not involve the 148 mechanical properties of the ECM but only the cell volume regulation system:

Selective-compression method
where Π is the osmotic pressure of ions in the culture medium and ≃ 0. approximations the amount of this compressive stress (the traction force applied by the matrix on 160 the cell) can be approximated as the applied osmotic pressure: In sharp contrast with the previous situation, for small dextran, the stress applied by the ECM on 162 the cell is tensile. In fact, the dominating effect is that small dextran compresses the cells but not 163 the ECM. Thus, cell compression is balanced by a tensile force in the ECM. This tension is given by where is the ECM shear modulus. Formulas (1), (2) and (3) hold in the ideal case, where osmolytes 165 do not interact with the matrix and the axisymmetric system has stress free boundaries at infinity 166 (the ECM ball radius is much larger than the cell radius). 167 In practice, for a moderate osmotic shock Π ≃ 5 kPa, the dextran concentration is much 168 smaller than the characteristic ion concentration of the external medium (few hundreds millimolars) 169 and the tension can be considered negligible: small ≃ 20Pa ≪ | big | ≃ 5kPa because the ECM is 170 soft. Therefore, in this condition, the presence of ECM makes the cell mechanically sensitive to a 171 moderate osmotic compression using big dextran molecules, but not when using small dextran 172 molecules. In both cases the cell volume is affected in the same negligible way, but the mechanical 173 stress applied by small dextran on the cell is negligible compared to that exerted by big dextran.    The present work therefore points at a direct mechanosensitive response of cells to the ECM 312 deformation. The microscopic structure of the ECM is modified under compression (e.g. density 313 increase and reduction of porosity), with consequences on the ECM rheology. Compression of the 314 ECM is clearly accompanied by an increase in its bulk modulus and, due to its fibrillar structure, to 315 a non-trivial and non-linear evolution of its stiffness (Sopher et al., 2018; Kurniawan et al., 2016). 316 For example, the rheological properties of synthetic ECM have been shown to affect growth of 317 aggregates and single cells through the regulation of streched-activated channels (Nam et al., 2019). 318 As integrin-dependent signals and focal adhesion assembly are regulated by the stress and strain   were also measured before and after buffer exchange. In the latter case, 50% of the culture medium 366 was simply aspirated and replaced by fresh medium not supplemented with dextran.  The matrix is a meshwork of biopolymers permeated by an aquaeous solution containing ions. These ions can also permeate the cell cytoplasm via specific channels and pumps integrated in the plasmic membrane (Hoffmann et al., 2009; Lang et al., 1998). For simplicity, we restrict our theoretical description to Na + , K + and Cl For simplicity we assume a spherical geometry with a cell of radius inside a matrix ball of radius . Each point in the space can therefore be localized by its radial position = where is radial unit vector. We assume a spherical symmetry of the problem such that all the introduced physical fields are independent of the angular coordinates and . Throughout this text, we restrict ourselves to a linear theory which typically holds when the deformation in the matrix is assumed to remain sufficiently small. A more quantitative theory would require to take into account both the non-linear aspects of the matrix deformation and the osmotic pressure created by the polymer.  (Staverman, 1952 ; Kedem and Katchalsky, 1958, 1963; Baranowski, 1991; Elmoazzen et al., 2009), assuming that the aquaeous solvent moves through specific and passive channels, the aquaporins (Day et al., 2014), we can express the incoming water flux in the cell at = as (Yi et al., 2003; Hui et al., 2014; Strange, 1993; Hoffmann  et al., 2009; Mori, 2012; Cadart et al., 2019): where Π , denote the osmotic pressures in the matrix phase and the cell while , are the hydrostatic pressures defined with respect to the external (i.e. atmospheric) pressure. The so-called filtration coefficient is related to the permeability of aquaporins. In a dilute approximation which we again assume for simplicity, the osmotic pressure is dominated by the small molecules in solution and thus reads where is the Boltzmann constant, the temperature, , , , and , are the (number) concentrations of sodium, potassium and chloride in the cytoplasm and the extra-cellular medium and is the extra-cellular Dextran (necessary small as big are excluded) concentration in the matrix phase. We neglect in (5) the osmotic contribution associated with the large macromolecules composing the cell organelles and the cytoskeleton compared to the ionic contributions. In a similar manner, the osmotic contribution of the matrix polymer is also neglected. At steady state, the water flux vanishes ( = 0) leading to the relation at = , Ions conservation. 516 As each ion travels through the plasma membrane via specific channels and pumps, the intensities of each ionic current at = is given by Nernst-Planck laws (Mori, 2012), where , , are the respective conductivities of ions, is the cell membrane potential, is the elementary charge and is the pumping rate associated to the Na-K pump on the membrane which is playing a fundamental role for cellular volume control (Hoffmann et al.,  2009). The factors 3 and 2 are related to the stochiometry of the sodium potassium pump.
Again, in steady state, currents , , = 0, leading to the Gibbs-Donnan equilibrium: where the active potentials related to the pumping activity , are = −3 ∕ and = 2 ∕ . Supposing that the cell membrane capacitance is vanishingly small (Mori, 2012), we can neglect the presence of surface charges and impose an electro-neutrality constraint for the intra-cellular medium: where is the average number of (negative in the physiological pH = 7.4 conditions) electric charges carried by macromolecules inside the cell and is their density. As macromolecules are trapped inside the cell membrane, we can express = ∕(4 3 ∕3) where is the number of macro-molecules which is fixed at short timescale and only increases slowly through synthesis as the amount of dry mass doubles during the cell cycle (Cadart et al.,  2019). Force balance. 544 At the interface between the cell and the matrix ( = ), we can express the mechanical balance as bulk + surf = . (10) In (10), bulk is the Cauchy stress in the cytoplasm which we decompose into bulk = skel − I, with a first contribution due to the cytoskeleton and a second contribution due to the hydrostatic pressure in the cytosol. The identity matrix is denoted I. The contribution due to the mechanical resistance of the cortex and membrane is denoted surf . In our spherical geometry, we can express surf = 2 ∕ where is a surface tension in the cell contour. Finally is the stress in the matrix phase for which we postulate a poro-elastic behavior such that, = el ( ) − I (the Biot coefficient (Coussy, 2004) is assumed to be one.) where is the Hooke's law with the (small) elastic strain in the matrix, the shear modulus and the drained bulk modulus. In the absence of cytoskeleton and external matrix, (10) reduces to Laplace law: and more generally reads, Such relation provides the hydrostatic pressure jump at the cell membrane ( = ) entering in the osmotic balance (6) and, combining (6) and (12), we obtain Assuming that the extra-cellular fluid follows a Darcy law, mass conservation of the incompressible water permeating the matrix can be expressed as where is the matrix porosity, the matrix permeability and the fluid viscosity. At steady state, = 0 and (14) is associated with no flux boundary conditions at and given by | , = 0.

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It follows that is homogeneous in the matrix and its value is imposed by a relation similar to (6) with an infinitely permeable membrane at : In (15), Π is the external osmotic pressure which reads where ( ) is the electro-static potential in the matrix.
Next, we again suppose for simplicity that the capacitance of both the porous matrix and the external media are vanishingly small leading to the electro-neutrality constraints where is the number of negative charges carried by the biopolymer chains forming the matrix and is their density. As we use uncharged Dextran, its concentration does not enter in expressions (18). Using, (17) in tandem with (18), we obtain Re-injecting this expression into (17), we obtain the steady state concentrations of ions in the matrix phase: Next, we make the realistic assumption that the chloride concentration (number of ions per unit volume) is much larger than the density of fixed charges carried by the polymer chains (number charges per unit volume): ∕ ≪ 1. Indeed using the rough estimates of Section A.4, the average number of charge carried per amino-acid is 0.06 and the typical concentration of matrix is 5 g/L. As the molar mass of an amino-acid is roughly 150g/mol, we can estimate in moles that ≃ 2mM while ≃ 100mM. We can thus simplify (20) up to first order to obtain, is imposed by Dextran since the ions only start to contribute to this difference at second order in the small parameter ∕ . We therefore conclude that, in good approximation, Π vanishes for small Dextran molecules that can permeate the matrix and equates to the imposed and known quantity for big Dextran molecules that cannot enter the matrix pores. Using the spherical symmetry of the problem, the only non vanishing components of the stress tensor are and = . Therefore, the local stress balance reads Assuming a small enough displacement, the non-vanishing components of the strain tensor are given by, = ∕ and = = ∕ where is the radial (and only non-vanishing) displacement component from an homogeneous reference configuration corresponding to a situation where the matrix is not subjected to any external loading and , = , . Using the poro-elastic constitutive behavior (11), satisfies This equation is supplemented with the traction free boundary condition at = = 0.
Combined with (23), the two above equations (24) and (25) lead to the solution where the introduced constants 0 is found using the displacement continuity at the cell matrix-interface: with given by the change of the cell radius from a reference configuration with radius . The general expression of therefore reads, ( ) = 2 4 3 + 3 3 + Π 3 3 − 3 leading to the following form of the total mechanical stress in the surrounding matrix: Combining (5) with (13) and taking into account (21), we obtain the relation linking the cell mechanics and the osmotic pressures inside the cell and outside the matrix: We suppose that the stress in the cytoskeleton is regulated at a homeostatic tension such that skel . def = Σ is a fixed given constant modeling the spontaneous cell contractility. We can then linearize the cell mechanical contributions close to = to obtain skel . + 2 =Σ − , whereΣ = Σ + 2 ∕ and the effective cell mechanical stiffness is = 2 ∕ 2 . We therefore finally get the linear relation, where,̃ In a similar way, we combine (8) with (9) with again (21) in the limit where ∕ ≪ 1 to express the electro-neutrality condition where we have additionally linearized the right handside close to = . The two equations (31) and (32) constitute our final model. 729 We begin by computing the cell radius and the cell membrane potential in the reference configuration where by definition = 0 and Π = = 0 as no Dextran is present at all. In this case, we solve for the membrane potential def = and radius in (31) and (32) to find their reference values. This computation strictly follows (Hoppensteadt and Peskin, 2012). Given that the typical concentration of chloride ions outside the cell is of the order of 100 milimolar, the osmotic pressure is of the order 10 5 Pa (i.e. an atmosphere). In sharp contrast, the typical mechanical stresses in the cytoskeleton and the cortex are of the order of 10 2 − 10 3 Pa (Julicher et al., 2007). Therefore the non-dimensional parameter is of the order of ∼ 10 −3 and will be neglected in the following. We then finally obtain the reference values, The pumping rate enables the cell to maintain a finite a volume. When → 0, → 1 and the cell swells to infinity because nothing balances the osmotic pressure due to the macromolecules trapped inside. So it is expected that dead cells will swell and lyse. The same happens if the pumping rate is to high. Indeed, as the membrane permeability of potassium is higher than the one of sodium, if the pumping rate is very high, a lot of potassium ions will be brought in (more than sodium ions will be expelled out) and to equilibrate osmolarity with the exterior, water will swell the cell until it bursts. Between these to unphysiological situations, computing the variation of volume with respect to the pumping rate, one gets that this variation vanishes when,

A.4 Cell volume in the reference situation
log .
At such pumping rate the volume is less sensitive to small variations in the pumping rate that may occur. Rough estimates. 769 The computation of the effective charge carried by macromolecules is complex. The folding of proteins and the electrostatic screening of charges between them (Manning effect) plays a role. See (Barrat and Joanny, 1997) for a review. We can still make a rough estimate in the following way. We assume that macromolecules are mostly proteins. At physiological pH = 7.4, three types of amino-acids carry a positive charge, Lysine (7%), Arginine (5.3%), Histidine (0.7%) while two others Aspartate (9.9%) and Glutamate (10.8%) carry a negative charge. Added to this, Histidine has a pKa = 6 smaller than the pH so the ratio of [histidine neutral base]/[histidine charged acid] is 10 pH−pKa = 25. Hence the contribution of histidine may be neglected. The occurrence of the aforementioned amino acids in the formation of proteins is also known. The average length of proteins is roughly 400 amino acids. We subsequently obtain the average effective number of negative charges as, = 400(9.9 + 10.8 − 7 − 5.3)∕100 = 25.
Such estimate needs to be refined and account for sugars and other macromolecules which carry more negative charges per chain but a interval from = 10 to = 100 charges is a plausible estimate. since the dynamical opening of channels due to some change in the membrane potential (Hodgkin and Huxley, 1952) as well as the mechanical opening mediated by membrane stretching can play a role and affect these quantities. Nevertheless a rough estimate can be given (Yi et al., 2003) = 2 × 10 −6 C.V −1 .s −1 and = 4.5 × 10 −5 C.V −1 .s −1 Also the pump rate is estimated in (Luo and Rudy, 1991), This pump rate is in good agreement with the optimal pump rate predicted by the model , This leads to an estimate of = 0.1. The density of macromolecules inside the cell is then found to be = 3 × 10 6 macromolecules per m −3 which is a correct order of magnitude (Milo, 2013). To further check the soundness of the above theory we can also compute the membrane potential and obtain = −73mV in good agreement with classical values . We use (31) and (32) to compute the ensuing small displacement . Assuming in good approximation that the osmotic pressure imposed by chloride ions is much larger (10 5 Pa) than the mechanical resistance of the cell cortex and the external matrix (10 3 Pa) ≫̃ we find that, Strinkingly, making the realistic simplifying assumptions that ≫ and ≫ , leads to the same displacement of the cell membrane in the two situations of small and big Dextran: showing that the two different osmotic loading are not distinguishable at that level. The main text relation (1) is obtained by assuming that the osmotic pressure of negatively charged ions is half the osmotic pressure of all ions. Since small ≪ Π by at least one order of magnitude, the most important feature that changes between small and big Dextran is that small > 0 while big < 0. The physical picture behind this is that small Dextran compresses the cell without draining the water out of the 21 of 32 matrix. Therefore, the cell behaves as a small inclusion which volume is reduced by the osmotic compression. In response, the matrix is elastically pulling back to balance the stress at the interface. In contrast, for big Dextran, the water is drained out of the matrix which therefore compresses the cell. Notice that, like the membrane displacement, the variation of the membrane potential def = − is the same in the two situations: where we have made the same previous simplifying assumptions that ≫̃ , ≫ and ≫ . Again such variation is negligibly small in our conditions where ≪ by several order of magnitudes. This further indicates that the biological response of the cell in response to a big Dextran compression has a mechanical rather than an electro-static origin. • the fresh medium is supplemented with big Dextran at Π = 5 kPa (Big, n = 18), 875 We observe that, when the MG is compressed by big dextran moelecules, the PA beads are also lose 25% of their volume, even though they are not directly in contact with the osmolytes (green). Such a volume loss is compatible with a mechanical pressure applied of the beads of few kPa (Dolega et al., 2017) and shows that the externally applied osmotic pressure results in a similar mechanical pressure applied on the beads through the drainage and compression of the MG meshwork. In contrast, if the dextran molecules are small enough to penetrate the MG and the PA beads, no mechanical stress is exerted on the inclusions (blue). The evolution of the rheological properties of ECM filling the interstitial space of MCS is very difficult to evaluate, as the interstitial layer is extremely thin (see previous section). However, we can empirically define an exclusion-size (i.e. porosity), above which globular molecules do not penetrate the gel. To evaluate this exclusion-size, we dip the MCS in a solution containing fluorescent tracers with different radii. As shown in Appendix figure 4a, tracers with = 4.4 nm and = 5.8 nm permeate the extracellular space of the MCS but not those larger than 14.8 nm. In order to quantify the relative amount of tracers inside the MCS, we compare the average fluorescence measured inside the MCS, ⟨ ⟩ and in the surrounding solution ⟨ ⟩. Appendix figure 4b reports the relative intensities ⟨ ⟩/⟨ ⟩ , obtained respectively at an external osmotic pressure Π = 0 Pa and at Π = 5 kPa. In both cases, the fluorescence level lowers with large tracers. From the results presented in this section, we deduce that: In order to estimate how cells react to an overall compression of the whole MCS, we measure the cell volume change within the aggregate. Cell contours are manually segmented from stack piles obtained with 2-photon imaging. Manual segmentation is performed with Amira software. Cell volume is extracted before and after application of osmotic pressure for the same cell in order the measure its compression. Spherical cells are excluded of the analysis as they may undergo cell division and display rapid volume changes. Cells larger than 7000 m 3 are also discarded from the analysis as they may be two cells rather than one. The results are reported in Appendix figure 5 and show that cells appear to be compressed both by small (40 kPa) and big (15 kPa) dextran molecules. The pressures are chosen to match with the experiments presented in figure 3 of the main article. Notice that this method is much less accurate than the "fluorescence exclusion" method used to determine the volume of individual cells. The results have to be taken as qualitative. In this section, we estimate the bulk modulus of the extracellular matrix. As interstitial ECM is difficult to characterize in-situ, we use matrigel (MG) beads to roughly estimate the rheological properties of ECM. Consistently with native ECM, large Dextran molecules were also excluded from microbeads made of MG suggesting an equivalent effective permeability (Dolega et al., 2020). To determine the bulk modulus of MG beads, we follow their compression at different dextran concentration. To facilitate the measurement, the beads are doped with fluorescent nanoparticles. In Appendix figure 6, we display the volume decrease ∕ 0 of MG beads as a function of the osmotic stresses, between 15 and 500 Pa ( 0 being the bead 25 of 32 volume before compression). The continuous line represents the best fit to a Mooney-Rivlin model, the derivative of which represent the bulk modulus (Rivlin and Saunders, 1951 The scope of this section is to illustrate how to determine cell motility inside an opaque multicellular aggregate. Previous works already indicate that pressure affects cell motility in multicellular spheroids, but the observations are limited to either the superficial layer (Alessandri et al., 2013) or to the long-term centripetal motion (Delarue et al., 2013). Recently, we developed a method to measure the cell velocity in the deep layers of MCS without using confocal microscopy, which is limited in terms of sample thickness and observation time. In our setup (Brunel et al., 2017, 2020) (Appendix figure 7a), the MCS is observed by phase contrast (Appendix figure 7c) and is simultaneously illuminated with an infrared laser (850 nm). The light scattered by the MCS in the forward direction produces an interference pattern, which is collected by a camera (Appendix figure 7b). From the of temporal fluctuations of this pattern (signal shown in Appendix figure 7d and its autocorrelation function in Appendix figure 7e), one computes the average velocity of cells, moving inside the MCS. It has to be noticed that this technique, an evolution of the Dynamic Light Scattering, provides information on the 3D motility, and not only on the 2D motion as previously measured by Alessandri et al. at the surface of MCS Alessandri et al. (2013). With this method, we measure the average speed in the three cases of interest: without pressure, when a pressure is selectively applied on the cells, but not on the ECM (small Dextran), and when the pressure is applied to the whole MCS (big Dextran). The results are shown in Appendix figure 7f and 7g: whereas the average speed is comparable in the first two cases (10±1 m/h; magenta and cyan), it is reduced by a factor of two when the compression is exerted on the entire MCS (4.8±0.5 m/h, blue). As a control, a subset of spheroids are exposed either to the drug alone, without dextran in solution or to Dimethyl Sulfoxide (DMSO). The MCS volumes are obtained by measuring the surface of their equatorial planes and considering them as a perfectly spherical object. The volume is measured before adding the drug, then 45 minutes after exposure to the drug alone (gray in Appendix figure 8) or to the drug supplemented with dextran (green in Appendix figure 8) and normalized to the initial volume of each MCS. We observe that drugs modify the MCS volume in different manners (see the figure below) as compared to the control (DMSO).This is in agreement with the fact such pharmacological perturbations are kwown to impact the single cell volume in different maners (Stewart et al., 2011). When dextran is added to the solution, the spheroids get compressed from their initial state (with the drug); such final compression (Dextran+Drug) is comparable to that obtained with dextran alone but the amout of compression with respect to the intial state varies depending on the drug. This result is compatible with our idea the 5 kPa Dextran compression reduces almost to the maximum the inter-cellular space and that cells are then almost fully connective in the final state. Thus, depeding on the amount of compression that the drug first creates, the ensuing compression with Dextran will change depending on the available inter-cellular space that remains. For instance, in the presence of cytochalasin, the extra-cellular space is already largely reduced compared to DMSO so when the osmotic compression follows, their is hardly no inter-cellular space which can still be compressed. This seems to be an additional indication that the MCS compressibility in response to a gentle osmotic pressure is more related to the rheology of the extracellular space than to the internal organization and contractility of the cytoskeleton as we argue in Dolega et al. (2020).