Chase-and-run dynamics in cell motility and the molecular rupture of interacting active elastic dimers

Cell migration in morphogenesis and cancer metastasis typically involves interplay between different cell types. We construct and study a minimal, one-dimensional model composed of two different motile cells with each cell represented as an active elastic dimer. The interaction between the two cells via cadherins is modeled as a spring that can rupture beyond a threshold force as it undergoes dynamic loading from the interacting motile cells. We obtain a phase diagram consisting of chase-and-run dynamics and clumping dynamics as a function of the stiffness of the interaction spring and the threshold force and, therefore, posit that active rupture, or rupture via active forces, is a mechanosensitive means to regulate dynamics between cells. Since the parameters in the model differentiate between N- and E-cadherins, we make predictions for the interactions between a placodelike cell and a neural crestlike cell in a microchannel as well as discuss how our results inform chase-and-run dynamics found in a group of placode cells interacting with a group of neural crest cells. In particular, an argument was made in the latter case that the feedback between cadherins and cell-substrate interaction via integrins was necessary to obtain the chase-and-run behavior. Based on our two-cell results, we argue that this feedback accentuates, but is not necessary for, the chase-and-run behavior.

Figure 1

Schematic representation of two cells (blue and green) with the red interaction spring containing $\text{N}$-cadherin molecules in parallel, each with spring stiffness $k_c$ and a filopod with spring stiffness $k_f$. Each blue and green spring represents the prominent stress fibers along the length of the cell and, therefore, is active with both the extended mode (top) and contracted mode (bottom). The blue filaments represent actin filaments, red rectangles, $\alpha$-actinin, and the green shapes, myosin minifilaments.

Figure 2

(a) Plot of the equilibrium spring length of the active spring and (b) of the friction coefficient at the leading edge of the cell as a function of the active spring extension. From Ref. [16].

Figure 3

(a) Plot of cell length $x_s=x_1-x_2$ as a function of time for the parameters specified above for $k_1=1\text{nN}/\mu \text{m}, x_{\text{eq}1}=50\mu \text{m}, x_{\text{eq}2}=5\mu \text{m}, l^{\downarrow }=46.5\mu \text{m}, l^{\uparrow }=48.5\mu \text{m}, {\gamma }_{11}=10\text{nN}\text{s}/\mu \text{m}, {\gamma }_{12}=20\text{nN}\text{s}/\mu \text{m}$, and ${\gamma }_2=20\text{nN}\text{s}/\mu \text{m}$. (b) Plot of position of the center of mass, $x_{s,\text{cm}}$, as a function of time for the same parameters. (c) Plot of the velocity of the center of mass as a function of time for the same parameters.

Figure 4

(a–c) The relative distance between the two cells, the center of mass position of the neural crest, and the center of mass position of the placode cell, all as a function of time. Here, $f_r=0.01\text{nN}$ and $k=5\text{nN}/\mu \text{m}$. The gray region in the top figure indicates when the interaction is tuned on. (d–f) The same as (a–c) but with $f_r=0.03\text{nN}$ and $k=5\text{nN}/\mu \text{m}$. (g–i) Here, $f_r=0.03\text{nN}$ and $k=5\text{nN}/\mu \text{m}$ but with larger friction coefficients for the PLL cell. We have set the active noise strength $A=0$ in these simulations.

Figure 5

The velocity of the center of mass for the NCL cell averaged over one cycle once the system has reached steady state as a function of rupture force with $k=5\text{nN}/\mu \text{m}$. To the left of the vertical dashed line is CAR dynamics; to the right, clumping dynamics.

Figure 6

Chase-run and clumping states for the two-cell model for the parameter values noted in the text. The symbols, blue circles (chase-run) and green triangles (clumping) indicate simulation data, while the corresponding blue and green shaded regions correspond to the analytical result. The units of $k$ are $\text{nN}/\mu \text{m}$ and the units of $f_r$ are $\text{nN}$.

Figure 7

Phase diagrams for the two-cell model as a function of $h$, with the chase-and-run state shown in blue (dark gray in grayscale) and clumped state shown in green (light gray in grayscale). Panels (a), (b), (c), and (d) correspond to $h=7, h=5, h=4$, and $h=3$, respectively. Parameter values used are the same as in the main manuscript except for $x_{\text{eq}2}$, which is changed to vary $h$. The units of $k$ are $\text{nN}/\mu \text{m}$ and the units of $f_r$ are $\text{nN}$.

Figure 8

Phase diagrams for the two-cell model as a function of $w$, with the chase-and run state shown in blue (dark gray in grayscale) and clumped state shown in green (light gray in grayscale). Panels (a), (b), and (c) correspond to $w=2, w=3$, and $w=4$, respectively. Parameter values used are the same as in the main manuscript except for $l^{\uparrow }$ which is varied to vary $w$. Again, the units of $k$ are $\text{nN}/\mu \text{m}$ and the units of $f_r$ are $\text{nN}$.

Figure 9

(a) The relative distance between the cells with and without noise with variance $A$ on each of the four beads. The curves with noise are typical curves. (b) The CAR-clumping phase boundary in the presence of noise, computed from the stochastic differential equation simulations.

Figure 10

(a) The relative distance between the two cells with and without feedback. The brown shading represents the presence of the interaction spring in the chase-and-run case, the gray, the clumping case. (b) Two cells moving apart from each other are not always able to rupture the interaction spring, i.e., escape. It depends on the rupture force. We have set the active noise strength $A=0$ in these simulations.