Contractile network models for adherent cells

Cells sense the geometry and stiffness of their adhesive environment by active contractility. For strong adhesion to flat substrates, two-dimensional contractile network models can be used to understand how force is distributed throughout the cell. Here we compare the shape and force distribution for different variants of such network models. In contrast to Hookean networks, cable networks reflect the asymmetric response of biopolymers to tension versus compression. For passive networks, contractility is modeled by a reduced resting length of the mechanical links. In actively contracting networks, a constant force couple is introduced into each link in order to model contraction by molecular motors. If combined with fixed adhesion sites, all network models lead to invaginated cell shapes, but only actively contracting cable networks lead to the circular arc morphology typical for strongly adhering cells. In this case, shape and force distribution are determined by local rather than global determinants and thus are suited to endow the cell with a robust sense of its environment. We also discuss nonlinear and adaptive linker mechanics as well as the relation to tissue shape.

Figure 1

Sketch of the system. (a) Side view of an adherent tissue cell. (b) Top view. The cell is assumed to be adherent at four discrete dots. Its contour shows an invaginated shape resulting from a balance between bulk forces directed toward the cell center and boundary forces directed along the cell periphery. (c) The actin cytoskeleton is tensed by myosin II minifilaments, which actively contract the network with forces ${\vec{F}}_{\mathrm{active}}$. If the network links are strained, restoring forces ${\vec{F}}_{\mathrm{mech}}$ appear. (d) Force-extension curve $F_{ij}\left(L_{ij}\right)$ of a link $ij$ in a simple Hookean network, a passive cable network, and an active cable network (in order of increasing dash lengths). ($k$ $=$ spring constant, $T$ $=$ motor stall force per length, $L_0$ $=$ initial link length, $L_c$ $=$ critical length).

Figure 2

Tension-free reference states. (a) Square network with link length $\ell =1$ and side length $d=10\ell$. (b) Triangular network with the same $\ell$ and $d$.

Figure 3

Tensed Hookean spring (HSN) and passive cable networks (PCN) with ${\tau }_H=0.2$. The color bar gives the dimensionless force $f$. (a) Contraction of a HSN in the square reference state. (b) Equilibrium shape of a HSN in triangular reference state. (c) PCN in square geometry. (d) PCN in triangular geometry. For the triangular reference shape, HSN (b) and PCN (d) differ in the amount of compression in the thin extensions (marked by arrows).

Figure 4

(a) Boundary line of the HSN or PCN for the square shape for network tension ${\tau }_H=0.2$ (lower set) and ${\tau }_H=0.6$ (upper set). The straight line shows the unconstrained reference shape, while the three curved lines represent different initial link lengths, namely $\ell =1,0.1,0.02$ (from bottom to top within one set). (b) Boundary line of the active cable network (ACN) for the square shape for $\tau ={10}^{-3}$ (lower set) and $\tau ={10}^{-2}$ (upper set). In this case, no unconstrained reference shape exists. The three lines again represent the initial link lengths $\ell =1,0.1,0.02$ (from top to bottom).

Figure 5

Contraction of an active cable network (ACN) with square shape. (a) and (b) Contraction of the network leads to arc formation. (c) With increasing tension tubes form near the adhesion points. Note the bundling of filaments at the edge. (d) For very large $\tau$, the network collapses onto the center tree. (e) The same situation as in (a) but with $\ell =0.5$. The contour is equal to that in (a) and the boundary forces are the same. (f) Square Voronoi network with 212 nodes and 316 links. Note the regular contour and the strong stress localization in the periphery. Tension values are [from (a) to (f)] $\tau ={10}^{-3},{10}^{-2},{10}^{-1},1,{10}^{-3},{10}^{-3}$.

Figure 6

Contraction of networks with square shape with an additional adhesion point in the middle of the bottom line. (a) HSN or PCN model with ${\tau }_H=0.2$. (b) ACN with $\tau ={10}^{-2}$. (c) Relative displacement $\delta$ of nodes with $x=5$ (vertical middle line) with and without the additional adhesion point at the bottom. $y$ is the node's $y$ coordinate in the initial network. The four top lines correspond to the HSN while the four bottom lines represent the ACN. Different symbols show different topologies (square=$\square$, diamond=$\diamond$, triangular=$△$, hexagonal=$◯$).

Figure 7

Arc fits. (a) Arc analysis for a tensed HSN or PCN. From bottom to top: ${\tau }_H=0.1$ ($+$), $0.2$ ($*$), $0.4$ ($\times$), $0.6$ ($◯$), $0.8$ ($\square$), 1 ($\diamond$). Symbols: Contour. Lines: Straight fits of the linear contour parts (the three bottom lines), least-squares fits of round contour parts to arcs with constant curvature (the three top lines). $*$ corresponds to the bottom line from the networks shown in Figs. 3a and 3c. (b) Contour analysis for the bottom line of an ACN with $\tau ={10}^{-3}$ ($\times$), ${10}^{-2}$ ($+$), $2.5\times {10}^{-2}$ ($*$), ${10}^{-1}$ ($◯$), $2.5\times {10}^{-1}$ ($\square$), $5\times {10}^{-1}$ ($△$) from bottom to top. Symbols denote node positions of the network arcs while the lines are least-squares fits of the contours to arcs of constant curvature. $\times$, $+$, $◯$, and dashed line correspond to the bottom lines from Figs. 5a, 5b, 5c, 5d.

Figure 8

Schematic representation of the two contour models. (a) In the tension-elasticity model (TEM), an isotropic surface tension $\sigma$ pulls the contour in along the normal direction, while the counteracting line tension $\lambda$ acts along the tangential direction. $r$ is arc radius, $L$ is contour length, and $d$ is spanning distance. (b) In the elastic catenary model, the inward pull is vertical and thus leads to an inhomogeneous line density of force along the elastic contour. $\phi$ denotes the tangential angle.

Figure 9

Relation of arc radii to internal tension of ACN for different dot distances and comparison to the tension-elasticity model (TEM). Symbols denote simulation results ($\square$ corresponds to square, $\diamond$ to diamond, $△$ to triangular, and $◯$ to hexagonal topology), the solid line is the numerical solution of Eq. (24), and the dashed line the analytical result Eq. (25). Side lengths are (from bottom to top) $d=10$, 19, 31, with critical tensions ${\sigma }_c\approx 0.114$, $0.060$, $0.036$.

Figure 10

Force distribution in adherent networks. (a) Force of vertical links which cross the straight line $y=5.5$ in the HSN from Fig. 3a (top) and the ACNs from Figs. 5a and 5b (bottom, middle). (b) Force in the bottom line links of an ACN with $\sigma ={10}^{-3}$ for different lattice constants (symbols). From top to bottom: $\ell =2\text{,}1\text{,}0.5\text{,}0.2$. The straight line gives the line tension obtained via the TEM; the curved line follows from Eq. (31). In both (a) and (b), on the $x$ axis we have the $x$ coordinate of the center of mass of the links. (c) Forces on adhesion dots exerted by the ACN (top) and HSN (bottom). Simulation results are shown as dots, while lines give the power-law fits. For the ACN we obtain $f=a{\tau }^b$ with $a\approx 6.16$ and $b\approx 0.783$ and for the HSN $f=a{\tau }_H^b$ with $a\approx 4.51$ and $b\approx 1.26$. (d) Power-law fits of adhesion dot force vs surface tension. ACN and elastic catenary results (top line, collapsed) and TEM (low line).

Figure 11

Contracted networks with wormlike chain (WLC) mechanics. (a) PCN with ${\tau }_H=0.47$. (b) PCN-WLC with ${\tau }_H=0.47$ and ${\ell }_n=0.5$. (c) ACN-WLC with $\tau =1$ and ${\ell }_n=0.5$. (d) Maximum invagination $h$ of linear (upper curve) and nonlinear (lower curve) ACN.

Figure 12

Arc fits for PCN and ACN with nonlinear links. (a) Contour of the PCN-WLC from Fig. 11b. Symbols belong to different values of ${\tau }_H$: $0.1$ ($+$), $0.2$ ($*$), $0.3$ ($\times$), $0.43$ ($\square$), $0.47$ ($\diamond$). $\diamond$ correspond to the bottom line of the network from Fig. 11b. (b) Contour of the ACN-WLC from Fig. 11c. Symbols are bottom line node positions, while the lines are circular fits. Motor force values are $\tau ={10}^{-2}$ ($\times$), $2\times {10}^{-2}$ ($+$), ${10}^{-1}$ ($*$), $2.5\times {10}^{-1}$ ($◯$), $5\times {10}^{-1}(\square$), 1 ($△$). Note that $△$ gives the bottom line from Fig. 11c.

Figure 13

Equilibrium shapes for adaptive networks. (a) Square HSN or PCN network with $ea\approx 9.40$ for the peripheral links and $ea\approx 1.75$ for the innermost links. (b) Square ACN network with $ea\approx 7.10$ at the periphery and $ea\approx 2.00$ for the internal links. Parameters are ${\left[ea\right]}_1=10$, $f_0=0.5$ [in (a)], ${\tau }_0={\tau }_1={10}^{-2}$, ${\left[ea\right]}_1=10$, $f_0=0.1$ [in (b)].

Figure 14

Contraction of an ACN anchored to dots of finite size. (a) Initial situation. Round tissue with radius $r_t$ adherent to 4 round dots of finite radius $r_d$ which form a square with side length $d_d$. (b) Contracted tissue. Parameters are $r_t=38$, $r_d=4$, $d_d=36$, $\tau =0.1$.