Sticking around: Cell adhesion patterning for energy minimization and substrate mechanosensing

Abstract

Tissue stiffness (Young's modulus) is a key control parameter in cell behavior and bioengineered gels where defined mechanical properties have become an essential part of the toolkit for interrogating mechanotransduction. Here, we show using a mechanical cell model that the effective substrate stiffness experienced by a cell depends, not just on the engineered mechanical properties of the substrate but critically also on the particular arrangement of adhesions between cell and substrate. In particular, we find that cells with different adhesion patterns can experience two different gel stiffnesses as equivalent and will generate the same mean cell deformations. In considering small patches of adhesion, which mimic focal adhesion complexes, we show how the experimentally observed focal adhesion growth and elongation on stiff substrates can be explained by energy considerations. Relatedly, energy arguments also provide a reason why nascent adhesions do not establish into focal adhesions on soft substrates, as has been commonly observed. Fewer and larger adhesions are predicted to be preferred over more and smaller, an effect enhanced by random spot placing with the simulations predicting qualitatively realistic cell shapes in this case.

Figure 1

Adhesive area changes the effective substrate stiffness experienced. (A) Schematic diagram of a circular cell with an adhesive ring. (B) Plot of deformation profile of a cell with $r_1/r_0=0.9,0.8,0.7,0$ (from bottom to top). $r_1/r_0=0$ corresponds to complete adhesion ($\gamma =7$). (C) Heatmap showing how mean cellular deformation varies with ring thickness (parameterized by $r_1$) and substrate stiffness (parameterized by γ). Along the contour line (black), cells display the same mean deformation. (D) Relative effective resistance plotted against internal ring radius ($r_1$) on substrates with $\gamma =\mathrm{5,10,15}$ (from bottom to top). (Here and in all further figures we set $P_0=0.7$ and $\nu =0.45$.) To see this figure in color, go online.

Figure 2

Arrangement of adhesion into localized spots facilitates localized regions of high deformation. (A) Schematic diagram of a circular cell with 10 spots evenly distributed around the edge. Heatmaps of the mean deformation on soft substrates with $\gamma =5$ (B–D) or stiff with $\gamma =11.34$ (E–G) with arrangements of 5 spots ($r_s=0.14r_0$) (B and E), 10 spots ($r_s=0.1r_0$) (C and F), and 20 spots ($r_s=0.07r_0$) (D and G) evenly distributed around the cell edge. Adhered area is maintained at 10% A, where A is the precontraction cell spread area (i.e., $A=\pi r_0^2$). Red arrows show the deformation of the midpoint of each spot. To see this figure in color, go online.

Figure 3

Increasing spot size increases apparent substrate stiffness: an effect which may be compensated for by the elongation of spots into elliptical patches. (A) Mean deformation plotted against adhered area for evenly distributed adhesive spots, with number of spots 5, 10, and 20 (from top to bottom). (B) Mean deformation plotted against spot aspect ratio for 10 evenly distributed spots with adhered area $A_{\text{ad}}=10\text{\%},12\text{\%},14\text{\%}$ (from top to bottom). Increasing the aspect ratio $b/a$ corresponds to an elongation toward the center of the cell. The black dotted line indicates $⟨\left|u\right|/r_0⟩=0.115$. (In (A) and (B) $\gamma =7$.) To see this figure in color, go online.

Figure 4

Focal adhesion growth reduces strain energy on stiff substrates but not on soft with spot elongation optimal. Heatmaps showing substrate strain energy against γ and the proportion of adhered area with 5 (A), 10 (B), and 20 (C) adhered spots evenly distributed around the cell edge. $W_S$ against adhered area for $\gamma =5$ (D) and $\gamma =15$ (E), as indicated by the dotted lines in (A–C), for 5, 10, and 20 spots. $W_S$ for an adhered ring plotted for comparison. $W_S$ against spot aspect ratio ($b/a$) for $\gamma =5$ (F) and $\gamma =15$ (G) for 10 spots at adhesion 5, 10, and 15% in blue, orange, and green, respectively. $W_S$ for an even distribution of 10 spots on substrates with $\gamma =5$ (H), $\gamma =7$ (I), and $\gamma =10$ (J). The blue line indicates circular spots of spot radius $r_s$. The orange line corresponds to elliptical spots with a fixed width but increasing length so that the aspect ratio increases as adhered area increases; here $W_S$ is plotted against the equivalent radius of circular spots. (Substrate strain energy is normalized by $h{E}_c{r}_0^2/\left(1-{\nu }^2\right)$.) To see this figure in color, go online.

Figure 5

Random placement of adhesion sites can generate apparently softer substrates compared with uniform placement and is energetically favorable. (A) Mean cellular deformation and (B) substrate strain energy plotted against the variance in angular gap size for spots restricted to the edge (blue pluses) and in an annular region ($0.6r_0<r<r_0$) (orange crosses). Results for an even distribution of spots are included for comparison at $\text{Var}\left({\theta }_g\right)=0$. (In each simulation there are 20 circular spots, covering 10% of the cell spread area, for the ring the same angular spot placements are chosen but the radial position is varied; $\gamma =7$.) Examples of the cell deformation observed in the above simulations. (C) corresponds to point in (A) with largest mean deformation; (D) is a corresponding ring distribution; (E) corresponds to the point in (A) with the least mean deformation; and (F) is a corresponding ring distribution. To see this figure in color, go online.