Figure 1.
Semiflexible region and actin gel in the model.
(A) Schematic representation of processes defining the model. The actin filaments (green) in the semiflexible region (SR) can bend, if they are sufficiently long. They exert forces on the leading edge membrane (blue line) and push it forward. They elongate by polymerization and shorten by attachment of cross-linkers (red dumbbells). Cross-linking also advances the gel boundary (red line) defined by a critical concentration of bound cross-linkers. Retrograde flow in the actin gel counteracts forward motion of the gel boundary. Filaments can also attach to the leading edge membrane and exert a pulling force. New filaments are nucleated from attached filaments. Filaments can get capped or severed and then vanish into the gel due to cross-linking or bundling of bent filaments. (B–D) Changes in SR structure during cycles of protrusion and retraction. (B) The formation of a transient lamellipodium is initiated by nucleation of single short filaments from actin bundles in the gel. (C) An actin network grows due to branching, the filament density in the SR increases and the leading edge protrudes. (D) As the filaments in the SR get longer, capping and severing rates increase, the filament density goes down and the lamellipodium retracts. While the SR depth stays narrow, filaments get so long that they have to bend and are likely to form arcs or bundles parallel to the leading edge.
Table 1.
List of model parameters and their values.
Figure 2.
Solutions of the model describing transient and stable lamellipodia.
(A–D) Simulations in the excitable regime with stationary filament density n = 0. At random time points (arrows, min, 5.1 min), the density of attached filaments is incremented by one, which corresponds to random nucleation of a filament from the cortex or from filaments oriented parallel to the leading edge. Random nucleation of one filament corresponds to a supercritical perturbation of an excitable system. The transient increase in filament density describes lamellipodium formation and collapse. (A, C, E) Density of attached (blue), detached (red) and capped (yellow) filaments and total filament density (black). (B, D, F) Filament length and SR depth (black). Attached (blue) and detached (red) filaments are almost equally long so that both lines overlap. (B, D) The SR depth remains constant as the filaments grow very long. Consequently, they have to bend and form arcs. (A, B) With the parameters from Table 1, (C, D) with
, all other parameters unchanged. Decreasing the capping rate has a similar effect as increasing the nucleation rate (see Fig. 3). Filament densities and duration of transients increase. The second increment does not lead to a transient lamellipodium formation in (C, D) because the filament density has not dropped below 1/µm yet and filament length is not decreased. (E, F)
, all other parameters as in A, B. The lamellipodium is stabilized since faster cross-linking prevents the filaments from getting too long and floppy.
Figure 3.
Maximum filament density of the transient lamellipodium in the excitable regime as a function of the nucleation rate.
(Dashed line) With and
as initial conditions. (Dotted line) With
and
as initial conditions.
,
, all other parameters as in Table 1. At
, a transition to a stable lamellipodium takes place. (Solid line) Value of the filament density of the stable fixed point existing above this. The leading edge protrusion velocity is proportional to the filament density because the gel cross-linking velocity is proportional to the filament density.
Figure 4.
Simulation of the measurements with epithelial cells from Burnette et [20].
The same simulation as in Fig. 2, fitted to the experimental data from Burnette et al. [20] (E, F). Random nucleation occurs more frequently, so that a new lamellipodium forms right after the collapse of the previous one. (A) Position of the leading edge (black) and the gel boundary (blue). The SR depth, which is the distance between leading edge and gel position, is shown as a black line in (D). (B) Velocities of the leading edge (black) and the gel boundary (light blue) and retrograde flow velocity (red). (C) Density of attached (blue), detached (red) and capped (yellow) filaments and total filament density (black). (D) Filament length and SR depth (black). Attached (blue) and detached (red) filaments are almost equally long so that both lines overlap. Parameters are ,
,
,
. Membrane tension is characterized by an external force
. All other values like in Table 1. (E) Experimentally measured leading edge position (Fig. 5b from [20]). (F) Measured leading edge velocity (Fig. 5a from [20]. E, F published with permission from Nature Cell Biology.
Figure 5.
Simulation of myosin inhibition.
The same simulation as in Fig. 4, but with a myosin contractility of 0.225 pN/µm2 instead of 5.555 pN/µm2. (A) Position of the leading edge (black) and the gel boundary (blue). (B) Velocities of the leading edge (black) and the gel boundary (light blue) and retrograde flow velocity (red). Comparison with Fig. 4 shows that the measured increase in period (C) is reproduced by our simulation. Filament densities, filament lengths and SR depth show basically the same behavior as without myosin inhibition, see Fig. 4 C, D. (C) The measured velocity map from Burnette et al. (Fig. 4 from [20]) shows that the period of protrusion cycles increases when myosin is inhibited by application of Blebbistatin. Published with permission from Nature Cell Biology.