Shape and Dynamics of Adhesive Cells: Mechanical Response of Open Systems

Cell adhesion is an essential biological process. However, previous theoretical and experimental studies ignore a key variable, the changes of cellular volume and pressure, during the dynamic adhesion process. Here, we treat cells as open systems and propose a theoretical framework to investigate how the exchange of water and ions with the environment affects the shape and dynamics of cells adhered between two adhesive surfaces. We show that adherent cells can be either stable (convex or concave) or unstable (spontaneous rupture or collapse) depending on the adhesion energy density, the cell size, the separation of two adhesive surfaces, and the stiffness of the flexible surface. Strikingly, we find that the unstable states vanish when cellular volume and pressure are constant. We further show that the detachments of convex and concave cells are very different. The mechanical response of adherent cells is mainly determined by the competition between the loading rate and the regulation of the cellular volume and pressure. Finally, we show that as an open system the detachment of adherent cells is also significantly influenced by the loading history. Thus, our findings reveal a major difference between living cells and nonliving materials.

Figure 1

Schematic of cells adhered symmetrically between an adhesive surface and a cantilever. Panels (a) and (b) show convex and concave cells observed in experiments (adapted from Ref. [15] with permission). (c) The deflection of the cantilever is $\delta =F{l}^3/3EI$, where $F$ is the force applied by the cell and $EI$ is the bending stiffness of the cantilever. Thus, the cantilever can be treated as a spring with a stiffness of $k=3EI/l^3$ and zero rest length. In (d) and (e), the cell shape is cylindrically symmetric and can be described by $r(s)$ and $z(s)$, where $s$ is the arc length. $\theta (s)$ is the tangential angle of the arc length, and ${\theta }_0$ is the contact angle. So the cell is convex when ${\theta }_0<90°$ and concave when ${\theta }_0>90°$. $r_a$ is the adhesion radius. $H$ is the cell height and $H_0$ is the separation of the adhesive surface and cantilever. $d(t)$ is the displacement of the fixed end of the cantilever. (f) The loading and unloading process.

Figure 2

Dynamic adhesion of cells adhered to an adhesive surface and a cantilever. (a) $r_0=18\mu \mathrm{m}$, $k=0.5\mathrm{N}/\mathrm{m}$; (b) $r_0=10.5\mu \mathrm{m}$, $k=0.5\mathrm{N}/\mathrm{m}$; (c) $r_0=10.5\mu \mathrm{m}$, $k=0.01\mathrm{N}/\mathrm{m}$. The parameters $H_0$ and ${\mathrm{\Gamma }}_0$ used here are marked by stars in Fig. 3. Other parameters are the same (Table S1 of Ref. [30]). The subplots show (I) external force $F$, (II) contact angle ${\theta }_0$, and (III) cell height $H$. The red, light blue, and light green curves represent the dynamic process of reaching the three stable states: (1) the cell is convex (${\theta }_0<90°$) and $F<0$; (2) the cell is convex and $F>0$; (3) the cell is concave (${\theta }_0>90°$) and $F>0$. The dark green and orange curves represent the spontaneous rupture and collapse of cells. The color of these curves corresponds to the color of the phase diagrams in Fig. 3. The dashed lines in subplots (I) and (II) indicate the lines of $F=0$ and ${\theta }_0=90°$, respectively.

Figure 3

Phase diagrams of cell shapes for various cantilever stiffnesses $k$ and cell sizes $r_0$. (a) $r_0=18\mu \mathrm{m}$, $k=0.5\mathrm{N}/\mathrm{m}$; (b) $r_0=10.5\mu \mathrm{m}$, $k=0.5\mathrm{N}/\mathrm{m}$; (c) $r_0=10.5\mu \mathrm{m}$, $k=0.005\mathrm{N}/\mathrm{m}$; (d) $r_0=10.5\mu \mathrm{m}$, $k=0.01\mathrm{N}/\mathrm{m}$. There are three stable regions (red, light blue, and light green regions) and two unstable regions (dark green and orange regions). The stars indicate $H_0$ and ${\mathrm{\Gamma }}_0$ used in Fig. 2. The blue circles in (b) and (d) are the theoretical predictions for the critical condition of the tension-induced rupture [30]. The inset in the orange region of (d) shows the collapse of the cell as ${\mathrm{\Gamma }}_0$ increases.

Figure 4

Dynamic detachment of convex [(a)–(f)] and concave [(g)–(l)] cells that are initially at steady state; the initial compression is $d_0=14\mu \mathrm{m}$ and $d_0=8\mu \mathrm{m}$ for convex and concave cells, respectively. Left: the shape evolution during the detachment. The external force [(a),(g)], cellular volume [(b),(h)], hydrostatic pressure difference [(c),(i)], adhesion radius [(d),(j)], membrane tension [(e),(k)], and contact angle [(f),(l)] versus the displacement $d(t)$ with various loading speeds $k_d$.

Figure 5

The dynamic detachment of adhesive cells depends on loading history. The time evolution of the (a) external force, (b) cellular volume, (c) adhesion radius, (d) membrane tension, (e) contact angle, and (f) hydrostatic pressure difference of convex cells. For all the curves, $d_0=17\mu \mathrm{m}$, $k_c=k_d=0.1\mu \mathrm{m}/\mathrm{s}$, but the waiting time is different ($t_w=0$, 400, and 800 s for the blue, green, and red curves). The black curves represent the loading stage.