Cell motility as an energy minimization process

The dynamics of active matter driven by interacting molecular motors has a nonpotential structure at the local scale. However, we show that there exists a quasipotential effectively describing the collective self-organization of the motors propelling a cell at a continuum active gel level. Such a model allows us to understand cell motility as an active phase transition problem between the static and motile steady-state configurations that minimize the quasipotential. In particular, both configurations can coexist in a metastable fashion and a small stochastic disorder in the gel is sufficient to trigger an intermittent cell dynamics where either static or motile phases are more probable, depending on which state is the global minimum of the quasipotential.

Figure 1

Three first bifurcations from the homogeneous state for $r=0$. (a) and (b) are bifurcation diagrams for the quasipotential and the cell velocity. They have a pitchfork supercritical structure. Black dots localize the bifurcation points. (c) and (d) show the profiles of $c$ and $w$ for some special points labeled with the corresponding colored circles on (a) and (b). Full lines correspond to locally stable branches or solutions while dashed lines are locally unstable.

Figure 2

Structure of the first bifurcation from the homogeneous state for $r=3$. (a) and (b) are bifurcation diagrams for the quasipotential and the cell velocity showing the subcritical nature of the bifurcation. The black dot localizes the first bifurcation point and the red dot the turning point. The thin dotted vertical lines represent the domain where both the static and motile configurations are locally stable. (c) and (d) show the profiles of $c$ and $w$ for some special points labeled with the corresponding colored circles in (a) and (b). Full lines correspond to locally stable branches or solutions while dashed lines are locally unstable.

Figure 3

Phase diagram in the parameter space $(\alpha ,r)$ characterizing the steady state of system (8). The black line is the locus of the first bifurcation point and the red line the one of the turning point along the first bifurcating branch (when it exists). The blue dashed line represents a Maxwell line. Above this line, the homogeneous solution is the global minimum of the Lyapunov functional $\mathcal{F}$ while, below this line, it is the nontrivial polarized solution. We use $exp\left(r\right)$ instead of $r$ to better graphically visualize the separation between the bifurcation, turning point, and Maxwell lines.

Figure 4

Effect of stochastic fluctuations on the cell metastable dynamics defined by system (15). (a) Probability densities of the distribution of velocity of a moving cell in four typical cases: In red the static configuration is the only steady state of the deterministic cell dynamics, in green both static and motile states are locally stable but the static state is the global minimum of the quasipotential, in blue the motile state becomes the global minimum and in black only the motile state is locally stable. (b) shows samples of the velocity dynamics in the four cases. Parameter $r=3$ and parameters defining the noise are $\mathrm{\Theta }=0.01$ and $e=0.001$. The simulations to obtain the probability densities start from the homogeneous distribution and are run over a nondimensional time of 1000. The transient state is removed and the distributions are symmetrized with respect to $V=0$ to minimize the computation cost.