Phenomenological modeling of durotaxis

Cells exhibit qualitatively different behaviors on substrates with different rigidities. The fact that cells are more polarized on the stiffer substrate motivates us to construct a two-dimensional cell with the distribution of focal adhesions dependent on substrate rigidities. This distribution affects the forces exerted by the cell and thereby determines its motion. Our model reproduces the experimental observation that the persistence time is higher on the stiffer substrate. This stiffness-dependent persistence will lead to durotaxis, the preference in moving towards stiffer substrates. This propensity is characterized by the durotaxis index first defined in experiments. We derive and validate a two-dimensional corresponding Fokker-Planck equation associated with our model. Our approach highlights the possible role of the focal adhesion arrangement in durotaxis.

Figure 1

Sketch of our model. Red circles represent focal adhesions. In our simulation, focal adhesions are randomly generated within an angular range bounded by red lines in the figure at each step. (a) For cells on a soft substrate, the distribution of FAs is relatively wide. (b) Conversely, for cells on a hard substrate, the distribution of FAs is relatively narrow.

Figure 2

Simulation result of uniform substrates and the variance of deflected angles. (a) Variance of deflected angles calculated by Monte Carlo sampling for ${\sigma }^2$ vs stiffness $k$, with $N=8$ and $r_i=1$ (red), $N=12$ and $r_i=1$ (blue), $N=12$ and $r_i\in (0.5,1.5)$ (yellow) for all $i$ and the fitted function $\sigma \left(k\right)$ with $\alpha =3.9$ and $\beta =-0.645$ [Eq. (3)] (purple). (b) Comparison between Monte Carlo sampling and direct simulation for $N=12$ and $r=1$ (red), with two lines overlapping as the purple line; for $N=12$ and $r_i\in (0.5,1.5)$, with two lines overlapping as the blue line; for $N=8$ and $r_i=1$, shown by the green line (direct simulation); and for $N=8$ and $r_i=1$, shown by the red line (Monte Carlo sampling). (c) and (d) Simulation of 1000 time steps with $v=1$ (arbitrary units). The initial position of 20 cells is $(0,0)$ and the initial moving direction is randomly selected. The black dots are the final positions of each cell. (c) On a soft substrate, $k=1$ and ${\theta }_i\in (-0.5\pi ,0.5\pi )$ for all $i$. (d) On a stiff substrate with $k=5$, the angle range is ${\theta }_i\in (-0.1\pi ,0.1\pi )$ for all $i$.

Figure 3

Direct simulation on the gradient matrix and DI. (a)–(c) Soft substrate $k=1$ in the left region and hard substrate $k=5$ in the right region. The central region has a constant gradient stiffness and varying width: (a) $L=100$, (b) $L=200$, and (c) $L=300$ (arbitrary units). (d) Durotaxis indices at several values of $L$; the index decreases as $L$ is increased. Every ten steps is counted as a big step.

Figure 4

The PDE solution. (a) Initial probability density function $p(x,\mathrm{\Phi };t=0)$. (b) Probability density function $p(x,\mathrm{\Phi };t=94)$ on a uniform substrate. Note that in this and the subsequent figure $\mathrm{\Phi }$ runs from $\pi /2$ to $5\pi /2$.

Figure 5

The PDE solution and comparison to simulation on a substrate with stiffness gradient. Probability density distribution $p(x,\mathrm{\Phi };t)$ at (a) $t=56$ and (b) $t=94$. (c) Full 2D density plot from direct simulation. (d) Comparison between direct simulations of 30 000 cells and the numerical solution of the Fokker-Planck equation for $\widehat{p}(x,t=94)$.

Figure 6

Direct simulations with random initial velocities and uniform initial spatial density.

Figure 7

Further test of the PDE. (a) Full PDE solution for periodic variation in stiffness and concomitant cell speed, with random initial velocities and uniform initial spatial density. (b) Comparison between direct simulations of 10 000 cells and the numerical solution of the Fokker-Planck equation for $\widehat{p}(x,t=200)$.