  Amoeboid cell migration is characterized by frequent changes of the direction
of motion and resembles a persistent random walk on long time scales. Although
it is well known that cell migration is typically driven by the actin
cytoskeleton, the cause of this migratory behavior remains poorly understood.
We analyze the spontaneous dynamics of actin assembly due to nucleation
promoting factors, where actin filaments lead to an inactivation of the
nucleators. We show that this system exhibits excitable dynamics and can
spontaneously generate waves, which we analyse in detail. By using a
phase-field approach, we show that these waves can generate cellular random
walks. We explore how the characteristics of these persistent random walks
depend on the parameters governing the actin-nucleator dynamics. In particular,
we find that the effective diffusion constant and the persistence time depend
strongly on the speed of filament assembly and the rate of nucleator
inactivation. Our findings point to a deterministic origin of the random walk
behavior and suggest that cells could adapt their migration pattern by
modifying the pool of available actin.
