Fig 1.
Schematic representation of the actin dynamics captured by Eqs (1)–(4).
Blue circles represent inactive nucleators. They are spontaneously activated at rate ω0, a process that is often associated with membrane binding. The activation rate is enhanced by already active nucleators, represented by green circles, which is captured by the parameter ω. Active nucleators generate new actin filaments (red) at rate α. The latter grow at velocity va and spontaneously disassemble at rate kd. Furthermore, actin filaments attract factors that inactivate nucleators. This complex process, which can involve several different proteins in a cell, is captured by the rate ωd.
Table 1.
Nondimensional parameter values used in this work unless indicated otherwise.
Fig 2.
Phase space diagrams for spatially homogenous dynamics.
A-C) Phase space for the FitzHugh-Nagumo Eqs (7) and (8) with a = 2, I = 0 (A), a = 0.04, I = 2 (B), and a = 0.4, I = 2 (C). D-F) Phase space for the dynamic Eqs (5) and (6) with kd = 5, α = 50 (D), kd = 50, α = 400 (E), and kd = 80, α = 400 (F). Other parameters as in Table 1. In each case, the nullclines are shown in red, the vector fields as blue arrowheads and an example trajectory in black. For the FHN equations, the diagrams show a bistable case (A), a limit cycle (B) and an excitable case (C). For Eqs (5) and (6) we present limit cycles (D, E) and an excitable case (F). For these equations, there is no bistable case.
Fig 3.
Snapshots of solutions for the actin concentration c to Eqs (1)–(4) in two dimensions with periodic boundary conditions.
A, B) Travelling planar waves for Da = 0.04, ωd = 0.28, va = 0.2 (A) and Da = 0.04, ωd = 0.32, va = 0.44 (B). Green arrows indicate the direction of motion. The disclinations in (B) might heal after very long times. C, D) Stationary Turing patterns for Da = 0.04, ωd = 0.45, va = 6.0 (C) and Da = 0.21, ωd = 0.42, va = 9.5 (D). For different initial conditions a pure hexagonal pattern of blobs can appear. All other parameters as in Table 1.
Fig 4.
Wavelength as a function of system parameters.
Orange dots represent values obtained from numerical solutions in two spatial dimensions with periodic boundary conditions (L = 1.0), blue lines are the results of a linear stability analysis, see Sect Linear stability analysis. Parameter values are ωd = 0.44 (A), ωd = 0.48 (B), va = 0.46 and ωd = 0.43 (C), va = 0.32 (D), va = 0.48 (E), and va = 0.46 and ωd = 0.43 (F). All other parameters as in Table 1.
Fig 5.
Shape and velocity of traveling waves in one dimension.
A) Actin and active nucleator concentrations c (green) and na (orange) for va = 0.8 and ωd = 0.35. Dots are from a numerical solution, solid lines are obtained from the variational ansatz Eq (23) and the solution Eq (39). B, C) Wave speed as a function of the actin polymerization velocity va (B) and the nucleator inactivation parameter ωd (C). All other parameters as in Table 1.
Fig 6.
Polymerization waves in presence of a phase field.
A) Schematic comparison of the discretized diffusion in absence (left) and presence (fight) of a phase-field, see Eq (29). B) Phase diagram of migration patterns as a function of the actin growth velocity va and nucleator inactivation parameter ωd. C-E) Example trajectories with cell outlines drawn at 8 equidistant points in time for va = 0.34 and ωd = 0.45 (diffusive migration, C), va = 0.22 and ωd = 0.38 (random walk with straight segments, D), and va = 0.46 and ωd = 0.43 (random walk with curved segments, E). Scale bars correspond to a length of 0.3. Other parameters as in Table 1.
Fig 7.
Effective parameters of random walk trajectories.
A-C) Diffusion constant D (A), speed v (B), and persistence time τ (C) as a function of the actin polymerization speed va. D-F) As (A-C), but as a function of the nucleator inactivation parameter ωd. Values were measured by fitting a persistent random walk model to the mean square displacement (MSD) of the respective trajectories. In (B) and (E), also the mean speed measured directly on the trajectories is shown (orange squares). Other parameters as in Table 1.