Figure 2. Cell shape and persistence of migration as a function of cell polarizability.

(A) Mean-squared displacement (MSD) for single-cell movements at different maximum cell polarity $\mathrm{\Delta }ϵ$ (stiffness parameters ${\kappa }_P=\mathrm{\hspace{0.17em}0.060}$, ${\kappa }_A=\mathrm{\hspace{0.17em}0.18}$; average polarization field ${ϵ}_0=\mathrm{\hspace{0.17em}225}$; signaling radius $R=\mathrm{\hspace{0.17em}5}$; cell-substrate dissipation $D=\mathrm{\hspace{0.17em}0}$; cell-substrate adhesion penalty $\phi =\mathrm{\hspace{0.17em}0}$; cytoskeletal update rate ${\displaystyle \mu =0.1}$; ${\displaystyle 100}$ independent simulations for each set of parameters). Single cells perform a persistent random walk, i.e. they move ballistically ($\text{MSD}\propto {\tau }^2$) for $\tau \ll {\tau }_p$, and diffusively ($\text{MSD}\propto \tau$) for $\tau \gg {\tau }_p$. Inset: Normalized velocity auto-correlation function for the same parameters as in the main figure. (B) Persistence time of directed cell migration plotted as a function of maximum cell polarity $\mathrm{\Delta }ϵ$, and perimeter stiffness ${\kappa }_P$ (area stiffness ${\kappa }_A=0.18$; average polarization field ${ϵ}_0=225$; signaling radius $R=5$; cell-substrate dissipation $D=\mathrm{\hspace{0.17em}0}$; cell-substrate adhesion penalty $\phi =\mathrm{\hspace{0.17em}0}$; cytoskeletal update rate ${\displaystyle \mu =0.1}$; ${\displaystyle 100}$ independent simulations for each set of parameters). The persistence time of the random walk increases with increasing cytoskeletal polarity and decreasing perimeter elasticity. (C) Cytoskeletal polarity also controls cell shapes, with crescent cell shapes (long persistence times) being observed at large cytoskeletal polartities, and round cell shapes (short persistence times) at small cytoskeletal polarities. Color code: cell polarization; cf. color bar in Figure 1C. (D) Single cell speed plotted as a function of maximum cell polarity $\mathrm{\Delta }ϵ$, and perimeter stiffness ${\kappa }_P$. (E) Single cell aspect ratio plotted as a function of maximum cell polarity $\mathrm{\Delta }ϵ$, and perimeter stiffness ${\kappa }_P$. (F) Speed and persistence time of single cells are correlated with the cell aspect ratio.

Figure 2—figure supplement 1. Role of substrate dissipation for cell shape and motility.

(A–D) Role of substrate dissipation for cells of varying maximum cell polarity $\mathrm{\Delta }\mathbf{}ϵ$. The aspect ratio ${\displaystyle l_{+}/l_{-}}$ (A), the speed $v$ (B), and the persistence time ${\tau }_p$ (C) as a function of substrate dissipation $D$ for a series of values for $\mathrm{\Delta }ϵ/{\kappa }_P$ indicated in the graphs. (D) Cell persistence and cell speed are correlated with the aspect ratio of the cell. Only cells with an aspect ratio larger than 2 are motile. In the simulations, the substrate dissipation was varied in the interval ${\displaystyle D\in [0,50]}$, and the maximum cell polarity $\mathrm{\Delta }ϵ\in \{20,30,40,50,60\}$. Fixed parameters: average polarization field ${ϵ}_0=225$; area elasticity ${\kappa }_A=0.18$; membrane elasticity ${\kappa }_P=0.060$; cytoskeletal update rate $\mu =0.1$; cell-substrate adhesion penalty ${\displaystyle \phi =0}$; ${\displaystyle 100}$ independent simulations for each set of parameters. (E–F) Role of substrate dissipation for cells of varying membrane stiffness ${\kappa }_P$. The aspect ratio ${\displaystyle l_{+}/l_{-}}$ (E), the speed $v$ (F), and the persistence time ${\tau }_p$ (G) as a function of substrate dissipation $D$ for a series of values for $\mathrm{\Delta }ϵ/{\kappa }_P$ indicated in the graphs. (H) Cell persistence and cell speed are correlated with the aspect ratio of the cell. Only cells with an aspect ratio larger than 2 are motile. In the simulations, the substrate dissipation was varied in the interval ${\displaystyle D\in [0,50]}$, and the membrane elasticity ${\kappa }_P\in \{0.054,0.057,0.060,0.063,0.066\}$. Fixed parameters: average polarization field ${ϵ}_0=225$; area elasticity ${\kappa }_A=0.18$; maximum cell polarity $\mathrm{\Delta }ϵ=50$; cytoskeletal update rate $\mu =0.1$; cell-substrate adhesion penalty ${\displaystyle \phi =0}$; ${\displaystyle 100}$ independent simulations for each set of parameters.

Figure 2—video 1. Single cell motility and shape for different maximum cell polarities (${\kappa }_P=0.060$, $R=5$).

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