Mapping cell cortex rheology to tissue rheology and vice versa

Étienne Moisdon, Pierre Seez, François Molino, Philippe Marcq, and Cyprien Gay

Phys. Rev. E 106, 034403 – Published 15 September 2022

Abstract

The mechanics of biological tissues mainly proceeds from the cell cortex rheology. A direct, explicit link between cortex rheology and tissue rheology remains lacking, yet would be instrumental in understanding how modulations of cortical mechanics may impact tissue mechanical behavior. Using an ordered geometry built on 3D hexagonal, incompressible cells, we build a mapping relating the cortical rheology to the monolayer tissue rheology. Our approach shows that the tissue low-frequency elastic modulus is proportional to the rest tension of the cortex, as expected from the physics of liquid foams as well as of tensegrity structures. A fractional visco-contractile cortex rheology is predicted to yield a high-frequency fractional visco-elastic monolayer rheology, where such a fractional behavior has been recently observed experimentally at each scale separately. In particular cases, the mapping may be inverted, allowing to derive from a given tissue rheology the underlying cortex rheology. Interestingly, applying the same approach to a 2D hexagonal tiling fails, which suggests that the 2D character of planar cell cortex-based models may be unsuitable to account for realistic monolayer rheologies. We provide quantitative predictions, amenable to experimental tests through standard perturbation assays of cortex constituents, and hope to foster new, challenging mechanical experiments on cell monolayers.

Received 22 April 2022
Revised 25 July 2022
Accepted 11 August 2022

DOI:https://doi.org/10.1103/PhysRevE.106.034403

Physics Subject Headings (PhySH)

Biomechanics Cell adhesion Cell assembly Cell mechanics Elastic deformation Elastic forces Mechanical deformation Rheological properties Rheology Shear deformation

Cell membrane Cytoskeleton Membrane structures

Microrheology Rheology techniques

Physics of Living SystemsPolymers & Soft Matter

Authors & Affiliations

Étienne Moisdon¹, Pierre Seez¹, François Molino², Philippe Marcq ³, and Cyprien Gay ^1,*

¹Laboratoire Matière et Systèmes Complexes, UMR 7057, CNRS and Université Paris Cité, 75205 Paris cedex 13, France
²Laboratoire Charles Coulomb, UMR 5221, CNRS and Université de Montpellier, Place Eugène Bataillon, F-34095 Montpellier, France
³PMMH, CNRS, ESPCI Paris, PSL University, Sorbonne Université, Université Paris Cité, F-75005 Paris, France

^*Corresponding author: cyprien.gay@univ-paris-diderot.fr

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Issue

Vol. 106, Iss. 3 — September 2022

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Images

Figure 1
Model ingredients: (a) Geometry: hexagonal cells tile the plane, with flat facets $a$ , $b$ , and $c$ . A representative, rectangular region of the monolayer is drawn in blue, with a volume $V = X Y Z$ . Relevant facets are labeled $a$ , $b$ (lateral) and $c$ (horizontal). Principal axes of the facets are drawn in red. (b) Rheological diagrams for the cortex corresponding to Eqs. (5) and (6), with the traceless and isotropic components of the stress. Here, as well as in Fig. 3, the (complex) shear modulus $g$ is represented by a parallelogram, the (complex) compression modulus $k$ by a circle, and the rest tension $σ_{0}$ by a filled disk.
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Figure 2
Views of the monolayer, force balance and macroscopic stress. (a) Top: top view. The facet lengths $ℓ_{a}$ and $ℓ_{b}$ and the orientation $θ$ of $b$ facets are shown. The repeating unit is shown as a dashed parallelogram. A simpler representative rectangular region, of same surface area $X Y$ , is shown in blue as in Fig. 1. Force balance between three neighboring cells: at the meeting point within the blue rectangle, all six cortex tensions add up vectorially to zero, which yields Eq. (16) (notations $σ_{a}^{y y}$ and $σ_{b}^{u u}$ have been simplified for clarity). Bottom: vertical cross section in the $x z$ plane. The force balance in the vertical direction implies that the pressure $p$ in the cytoplasm integrated over the whole cell width $X$ balances the tensions $σ_{a}^{z z}$ on either side of the cell, as expressed by Eq. (17). (b) Macroscopic stress. The $y y$ component of the in-plane macroscopic stress, $Σ_{y y}$ , can be expressed by counting all forces that are exerted across a perpendicular plane $P_{y y}$ , represented as a dashed horizontal line. Such forces are readily enumerated from the monolayer cross section in the $x z$ plane shown below the top view. They include the pressure $p$ in the cell (integrated over the monolayer thickness, $Z$ ), the tension $σ_{c}^{y y}$ of both horizontal layers, as well as the tensions $σ_{a}^{y y}$ of the relevant lateral facets, integrated over the facet height $Z$ . This provides the expression of Eq. (30). The expression of the $x x$ component $Σ_{x x}$ , provided by Eq. (29), is obtained in a similar way (see monolayer cross section in the $y z$ plane on the right-hand side), except that the lateral facet tensions, $σ_{b}^{u u}$ , must be projected onto the $x$ axis (angle $θ$ ; see also Fig. 1).
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Figure 3
Rheological diagrams for the monolayer. [(a), (b)] The monolayer compression modulus $K$ and shear modulus $G$ can be viewed as a mechanical arrangement of cortex properties. The symbols are those of Fig. 1. Note that cortical tensions $σ_{0}$ in Fig. 1 become springs in the present figure (with identical physical units). The coefficients are those of Eqs. (35) and (36), respectively. $Ψ$ is defined by Eq. (2). Indices $H$ and $L$ refer to horizontal vs lateral facets. (c) The monolayer Young modulus $E$ can be viewed as equivalent to four times its compression modulus $K$ in series with four times its shear modulus $G$ .
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Figure 4
Experimental data: cortex rheology. (a) Experimental data points from [22] (mN/m), represented by the red symbols and their error bars, are consistent with Eq. (49) (contractile visco-fractional cortex rheology), with the parameter values: $\hat{c} = 60 mN s^{B} / m$ , $\hat{η} = 250 mN s / m$ , $\hat{β} = 0.19$ (black lines). We observe a low-frequency viscous limit, and a high-frequency fractional limit, with exponent $B$ , dominated by the conservation modulus. (b) Red symbols and their error bars are obtained from data plotted in panel (a) using the mapping (35, 36, 37, 38) with $Ψ = 1$ , $σ_{0 H} = σ_{0 L} = 0.3 mN m^{- 1}$ , $ν_{c} = 0.41$ , and $g = (1 - ν_{c}) k / (1 + ν_{c})$ (blue lines). They are consistent with Eq. (48) (tissue viscoelastic fractional rheology), with the parameter values: $c_{β} = 500 mN s^{B} / m$ , $η = 170 mN s / m$ , $β = 0.19$ , $K_{0} = 21 mN / m$ . We observe a low-frequency elastic limit, and a high-frequency fractional limit, with exponent $β$ , dominated by the conservation modulus. Dashed line: conservation modulus; dotted line: loss modulus.
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Figure 5
Tissue moduli based on a Maxwell cortex rheology. Dashed lines: conservation moduli; dotted lines: loss moduli (mN/m). (a) Cortex rheology, Eq. (57), with parameters $η = 30 mN s / m$ and $g = 30 mN / m$ . (b) Tissue rheology, obtained using Eqs. (35) and (36) with parameters $ν = 0.4$ , $Ψ = 1$ and $σ_{0} = 0.3 mN / m$ . We obtain a low-frequency (not shown) and a high-frequency elastic limit, with an intermediate dissipative regime.
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Figure 6
Deformation modes. [(a)–(d)] The monolayer is stretched in the $x$ direction. [(e)–(g)] The monolayer is stretched isotropically in the $x − y$ plane. [(a), (e)] The blue dashed rectangle with intermediate lines is a guide for the eye that matches the undeformed (dashed) hexagon. The corresponding rectangle after deformation is drawn with a blue, solid line. [(b), (f)] The deformation is affine (hexagon deformation matches overall deformation, see intermediate blue lines, which implies $ε_{a} = ε_{y}$ ), when (b) $N = 0$ or when (f) the deformation is isotropic. (c) The facets in (a) shrink more ( $ε_{a} < ε_{y} < 0$ , $N < 0$ ) than the overall sample (compare deformed hexagon and intermediate blue lines). (d) The facets in (a) shrink less ( $ε_{y} < ε_{a} \leq 0$ , $N > 0$ ) than the overall sample (compare deformed hexagon and intermediate blue lines). (g) Under isotropic in-plane stretching, while horizontal facets are stretched isotropically, lateral facets are elongated along their respective horizontal directions in the $x − y$ plane and shrink along the direction $z$ .
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