Physics of adherent cells

One of the most unique physical features of cell adhesion to external surfaces is the active generation of mechanical force at the cell-material interface. This includes pulling forces generated by contractile polymer bundles and networks, and pushing forces generated by the polymerization of polymer networks. These forces are transmitted to the substrate mainly by focal adhesions, which are large, yet highly dynamic adhesion clusters. Tissue cells use these forces to sense the physical properties of their environment and to communicate with each other. The effect of forces is intricately linked to the material properties of cells and their physical environment. Here a review is given of recent progress in our understanding of the role of forces in cell adhesion from the viewpoint of theoretical soft matter physics and in close relation to the relevant experiments.

Figure 1

Passive bulk soft matter examples that are important model systems for the understanding of the material properties of cells and tissues. (a) Liquid crystals often form nematic phases, with no positional but orientational order. (b) In a dilute polymer solution, each polymer forms a globule that is well separated from the other polymers. (c) A cross-linked polymer gel can behave like an elastic solid but with much weaker rigidity. (d) Lipids self-assemble into fluid bilayers that at high concentration in turn tend to self-assemble into stacks, the so-called lamellar phase.

Figure 2

Simple soft matter models relevant to the passive features of cell adhesion to a flat substrate. (a) A liquid droplet adhering to a surface is governed by surface tension. (b) A solid elastic sphere gains adhesion energy by forming a contact region whose size is determined by the balance of the adhesion and shear deformation energies. (c) A closed shell of a fluid, lipid bilayer (vesicle) is governed by bending energy. (d) A polymeric capsule has both bending and stretching energy; their interplay can lead to buckling in the contact area.

Figure 3

Schematic drawing of an animal cell in suspension. Such a cell is essentially round due to its effective surface tension. Important cellular organelles responsible for its internal structure and mechanical properties include (1) the plasma membrane, a lipid bilayer that envelopes the entire cell and carries different proteins, including transmembrane receptors; (2) other membrane structures (thin lines) such as the two membranes around the nucleus containing the genes, the endoplasmic reticulum, the Golgi apparatus, and different kinds of vesicles; (3) the actin cortex, a thin shell comprising a polymer network underlying the plasma membrane; and (4) the microtubule system (thick lines), a system of relatively stiff polymers that radiate outward from the microtubule organizing center that is attached to the nucleus.

Figure 4

Schematic drawing of an adherent animal cell. Such a cell typically has the shape of a fried egg, with the nuclear region protruding in the middle while the rest of the cell remains relatively flat. As opposed to Fig. 3 for a freely suspended cell, here we do not depict the membrane or microtubule systems. In addition to the actin cortex, the actin cytoskeleton now forms several additional subsystems that extend throughout the cytoplasm. We do depict a dendritic actin networks that pushes outward against the plasma membrane (lamellipodium) and contractile actin filament bundles (stress fibers) that are anchored to the cellular environment through transmembrane receptors that bind extracellullar ligands.

Figure 5

Scheme for the overall force balance in an adherent cell. There are two actin-based processes that contribute to force generation at the cell-material interface. Contraction by myosin II motors in actin networks and bundles corresponds to a stretched spring pulling inward in the cell center. Lamellipodium growth against the membrane corresponds to a compressed spring pushing outward at the cell periphery. The entire system is constrained by the cell envelope. Because of the position of the adhesion sites, a contractile force dipole emerges as the effective traction pattern on the substrate. This leads to deformation of the substrate (compression below the cell body, and elongation away from the cell).

Figure 6

Spatial organization of focal adhesion growth and the actin cytoskeleton. The actin lamellipodium (LP) is assembled at the leading edge and flows from there toward the cell center. Small adhesions are formed along the way and mature into focal adhesion as they move with the actin flow. At the boundary with the myosin-dominated lamella (LM), only a few mature focal adhesions persist; those are stabilized by large contractile forces which are mainly due to the activity of myosin II minifilaments in stress fibers.

Figure 7

Schematic view of a focal adhesion. The transmembrane adhesion receptors from the integrin family (a heterodimer with two subunits) bind to the extracellular matrix (for example, collagen) on the outside and are cross-linked by cytoplasmic proteins such as talin in the inside. Talin binds to actin and this binding is further strengthened by proteins such as vinculin. The contractility of the actin cytoskeleton is determined by the activity of myosin II minifilaments.

Figure 8

Minimal model for an adhesion cluster under force: $N_t$ adhesion receptors are arranged along the membrane. Because of this geometry, they share the load. At any time, $N(t)$ bonds are closed and $N_t-N(t)$ bonds are open. The applied force $F$ is equally distributed over the closed bonds due to the parallel architecture $F_{\mathrm{bond}}=F/N(t)$. A bond dissociates with a force-dependent unbinding rate $k_{\mathrm{off}}(F_{\mathrm{bond}})$. The molecule can rebind with a constant rate $k_{\mathrm{on}}$. Adapted from 98.

Figure 9

Average cluster lifetime $T$ for $N_t=1$, 2, 5, 10, 15, and 25 (from bottom to top) as a function of $F/F_b{N}_t$ for $k_{\mathrm{on}}=k_0$ as follows from Eq. (26). The vertical line is the critical force predicted by Eq. (23) and the dashed line is the lifetime predicted by Eq. (22). Adapted from 99.

Figure 10

Selected trajectories simulated with the stochastic master equation model. Rebinding rate $\gamma =1$. (a) Force below threshold $f/N_t=0.25$. Small clusters ($N_t=10$ and 100) are unstable due to fluctuations. (b) Force above threshold $f/N_t=0.3$. Now all cluster sizes are unstable. Lines are first moments. Adapted from 98.

Figure 11

Minimal model for an adhesion cluster that bridges two surfaces that move relative to each other with velocity $v$. Here we discuss this case in the limit of continuous ligand and receptor coverage. The situation described here is very similar to sliding friction as usually studied for macroscopic objects.

Figure 12

Biphasic relation between dimensionless flow $V$ and dimensionless traction force $f_t$ predicted by the minimal model for cluster size $N_t=25$ and for different values for the rebinding rate. Symbols are the results of stochastic simulations with $N_t$ single bonds. Adapted from 294.

Figure 13

The nonlinear relation between the flow velocity $v$ and the traction force $F_T$ from Fig. 12 leads to a region of bistability for the flow velocity $v$ as a function of the driving force $F_D$. Parameters $N_t=25$, $\gamma =10$, and $\xi k_0/\kappa =0.03$.

Figure 14

Model of an adhesion cluster loaded by equal and opposite forces of magnitude $F$, each applied to an elastic half-space. $E_C$ and ${\nu }_C$ denote the Young’s modulus and Poisson’s ratio of the cell, and $E_S$ and ${\nu }_S$ denote those of the substrate. $2a$ and $b$ are the linear dimensions of the adhesion cluster and the distance between receptor-ligand bonds, respectively.

Figure 15

Calculated velocities of the front (closer to the direction of the pulling force in this figure, to the right) and back of the adhesion for a rigid matrix. Courtesy of A. Nicolas and adapted from 24 and 247.

Figure 16

Cell shapes are often dominated by tension. (a) Single cells in solution are usually round. (b) Isolated tissue cells like fibroblasts commonly show an invaginated shape between distinct sites of adhesion. (c) Epithelial cells in closely packed tissue are usually polygonal, in both two and three dimensions (here a polyhedron is schematically depicted for the 3D case). (d) Cylindrical cell extensions such as axons tend to pearl when the tension is increased.

Figure 17

Simple tension model for cell shape on a flat substrate. (a) The cell is very flat and therefore effectively two dimensional. The outlined region is shown in (b) with more detail. (b) Along the contour between two neighboring adhesion sites, surface tension $\sigma$ pulls inward, while line tension $\lambda$ pulls tangentially. (c) Along the contour, surface tension and line tension pull in normal and tangential directions, respectively. For a circular arc, radius $R$, contour length $L$, and spanning distance $d$ are geometrically related to each other. (d) For a house-shaped cell, all arcs have the same radius, although the spanning distance is larger on the diagonals. Also shown are the traction forces derived from the shape model. Adapted from 30 and 132.

Figure 18

(a) House-shaped cell on an adhesive micropattern created by microcontact printing. Note the circular arcs with radii that are larger at the two diagonals. From Fig. 1(B) from 28. (b) Cell on a pillar array that allows a simple readout of local forces from measurements of the deflection of each pillar. Circular arcs describe most of the cell contour, but not, for example, at the upper left corner, where an internal fiber distorts the contour. From 30.

Figure 19

Two model approaches for whole cells. (a) Cellular Potts models use spins on a lattice to simulate the contour between the cell and its surroundings. From 364. (b) Finite element methods (FEM) can be used to implement any material law of interest. Contractility is implemented by thermoelasticity. The vector field represents the direction and activation level of stress fibers. From 82.

Figure 20

(a) A spring network which is constrained to be located at the four adhesion points indicated (and in which the equilibrium spring length is smaller than the distance between neighboring adhesion points divided by the number of springs along that line), relaxes to a shape similar to the FEM model—that is, one with relatively flat free edges. Note also the variation in force along the boundary indicated by the colors. (b) An actively contracting cable network results in circular arcs and shows hardly any variation in force along the boundary. (c) The results do not depend on network topology as shown here for a disordered network. (d) Arc radius $R$ depends on network tension $\sigma$ as predicted by the tension-elasticity model. Adapted from 132.

Figure 21

Two-spring model for cell-substrate interactions. In addition to springs that characterize the cellular and substrate deformations, the cell exerts active, contractile forces shown by the double arrows. This simple cartoon shows that cells can measure the stiffness of their environment, which is, however, convoluted with their own stiffness. In contrast to Fig. 5, here we do not consider compression of the substrate, because we consider only the far field.

Figure 22

Simple model of a contractile and widely spread cell on an semi-infinite elastic substrate.

Figure 23

(a) The experimental values of the stress-fiber order parameter $S\sim {\overline{P}}_1$ (which indicates the extent to which the nascent stress fibers are aligned along the long axis of the cell) for three groups of stem cells (of aspect ratios 1.5, 2.5, and 3.5) as a function of the Young’s modulus of the matrix $E_m$. For the smallest aspect ratio, the order is clearly maximal for $E_m\sim 11\mathrm{kPa}$; for the other aspect ratios this trend may also be obeyed although it is less clear. The fit is motivated by the theory described in the text. From 395. (b) Aspect ratio of stem cells as a function of substrate elasticity. The point marked blebb refers to cells where myosin activity has been suppressed by treatment with the drug blebbistatin; this shows that the peak observed for untreated cells is related to cell contractility. From 283.

Figure 24

Phase diagram for positionally disordered cells. At low values of the scaled cell density $\rho$, an orientationally disordered (paraelastic) phase ($p$) prevails. At high cell density, orientational order sets in, with a nematic stringlike (ferroelastic) phase ($f$) at low values of Poisson ratio $\nu$, and an isotropic ringlike (antiferroelastic) phase ($af$) at large values. From 33.

Figure 25

Schematic view of two striated fibers. Striated stress fiberlike actomyosin fibers form close to the cell-substrate interface of adherent, nonmuscle cells. Each fiber is a bundle of aligned actin filaments that has a sarcomeric subarchitecture: $Z$ bodies (containing $\alpha$-actinin) that cross-link actin filament barbed ends alternate with regions rich in myosin II in a periodic fashion. Striated fibers can slide past each other until their periodic structures are in phase. Adapted from 115.