Extra-cellular Matrix in cell aggregates is a proxy to mechanically control 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 cells within multicellular aggregates with small osmolites and global aggregate compressions with big osmolites, 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 but, by draining the water out of the extracellular matrix, the global one imposes a residual compressive mechanical stress on the cells while the selective one does not. We conclude that, in aggregates, the extracellular matrix is as a sensor which mechanically regulates cell proliferation and migration in a 3D environment.


25
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, necessary for morphogenesis and homeostasis (Vogel, 2018).  ing 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  using osmolytes with gyration radii respectively larger and smaller that the ECM pore sizes (fig-89 ure 2). As osmolytes, we use dextran molecules ranging from 10 to 2000 kDa. As a proxy of ECM,

100
The small amount of compression, which we neglect, can be addressed by thermodynamic the-101 ories involving an interaction between the matrix and the permeating polymer (Brochard, 1981;102 Bastide et al., 1981). Conversely, big dextran molecules occasion a large compression of up to 103 63±5% of the initial volume (quantification in figure 2b).

104
Analogous experiments are performed using individual cells and multicellular spheroids. In-105 terestingly, the volume loss of individual cells is not measurable up to 10 kPa, and becomes ap-106 preciable at 15 kPa, with a relative compression Δ ∕ = 5 ± 5% (figure 2c properties of the ECM but only the cell volume regulation system: where Π is the osmotic pressure of ions in the culture medium and ≃ 0. 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 139 the cell is tensile because the dominating effect is that small dextran does not compress the ECM 140 but acts on the cell which compression is balanced by a tensile force in the ECM. This tension is 141 given by To test our theoretical predictions, we follow the evolution of the interstitial space inside MCS sub-153 mitted to osmotic compression, occasioned either by small or big dextran. To improve the optical 154 performance and to measure changes in the extracellular space after compression by small and 155 big dextran, we inject the spheroids (4-5 days old) into a 2D confiner microsystem (figure 3a). MCS 156 are confined in the microsystem for two to five hours to relax, the medium being supplemented

307
Beads are eventually transferred to PBS phase by washing out the surfactant phase.   Our aim is to qualitatively understand the nature of the steady state mechanical stress and displacement of a cell nested in a matrix in two paradigmatic situations: 540 541 • when some small osmolites (typically dextran) that can permeate the matrix pores are introduced in the solution, 542 543 • when some big osmolites that are excluded from the matrix are introduced in the solution.

545
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. From Kedem-Katchalsky theory (Staverman, 1952;Kedem andKatchalsky, 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 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 Π = ( + + + ) and 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.

598
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) 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.
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, 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. In steady state, = 0 and (14) is associated with no flux boundary conditions at and given by 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 , and denote the ions concentrations in the external solution and the concentration of Dextran added to the external solution. Ions conservation.

679
As we are interested about the steady-state only, the Poisson-Nernst fluxes of ions concentrations in the matrix locally vanish leading to: 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 obtain that ≃ 2mM while ≃ 100mM. We can thus simplify (20) up to first order to obtain, As a result, we obtain that the only steady state contribution of 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.
It then follows from (15) that the hydrostatic pressure equilibrates with the imposed osmotic pressure, Force balance.

732
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 an 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 . Using and (23) and (29)  .
In the limit where ≫ ,̃ = − + 4 3 and̃ = 1 + 4 3 . Next, using (8) and (21) and neglecting ∕ ≪ 1 we obtain the relation linking the externally controlled osmolarity with the cell and matrix mechanics: 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.

808
A.4 Cell volume in the reference situation 809 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 can balance 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 come in until the cell bursts because of the potassium ions pumped inside. Between these to unphysiological situations, computing the variation of volume with respect to the pumping rate, one gets that this variation vanishes when, log .
At such pumping rate the volume is less sensitive to small variations in the pumping rate that may occur. Rough estimates.

849
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 very 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, Arginine, Histidine while two others Aspartate and Glutamate 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 charges but a interval from = 10 to = 100 charges seems reasonable. Estimate of requires the knowledge of physiological external concentration of ions = 150mM, = 140mM and = 10mM as well as conductances of sodium and potassium ions through the plasmic membrane. Here again the situation is complicated 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), = 2.78 × 10 −12 mol.s −1 .
This pump rate is in good agreement with the optimal pump rate predicted by the model , 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 .  891 We now consider the case where, from the reference configuration presented in the previous section we impose an additional osmotic pressure in the external solution with Dextran polymers Π = . We recall that according to formula (22), Π = 0 for small Dextran molecules while Π = Π for big Dextran molecules. 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, = − ( ) 1∕3 ((̃ − 1)Π + Π )