Figure 1

An illustration of the interactions between the membranal proteins and the membrane shape, as described in the model. The squares (circles) represent the concentration of nucleators of branched (bundled) actin ($c_h$, $c_v$), respectively; $h$ is the height deflection of the membrane. We illustrate a molecular motor (myosin) that carries a cargo that enhances actin bundling at the membrane. Such motors transport cargo along the bundled filaments, and not on the branched network. One example is the myosin-IIIa which transports espin cargo (actin bundling protein) to the filament tip [28].Reuse & Permissions

Figure 2

Dispersion relations showing the linear instability of a uniform state to finite wave number perturbations. Parameters are given in Table .Reuse & Permissions

Figure 3

Asymptotic localized solution for $S_{v2}=0.32$ and $C_p^{\prime }=0.005$ while other parameters as in Table .Reuse & Permissions

Figure 4

(a) Qualitative numerical representation of solutions to Eqs. (3) on periodic domains in the absence of actin-membrane coupling, i.e., $C_p=C_p^{\prime }=0$. Black arrows depict the direction of front motion, i.e., either expansion or contraction with respect to the bottom uniform state, while the solid/dashed profiles correspond to two different times of the evolution and the horizontal dashed lines denote uniform solutions. For an expanding process $t\text{(solid}\text{line})<t(\text{dashed}\text{line})$, the initial state ($t=0$) was a local perturbation located at the center of the domain. For a contracting process the times are $t\left(\text{solid}\text{line}\right)>t\left(\text{dashed}\text{line}\right)$ and the initial local perturbation was located at the domain ends. Asymptotically the system converges to a uniform state. (b) Numerical representation as in (a) but with $C_p\ne 0$. Here the system converges to a localized steady-state solution (LS). The edges of the LS are marked by the shaded box, and have a width $L_f$. (c), (d) Numerical space-time plots where contour levels correspond to normalized membrane height. In (c) the initial perturbation is below the threshold and eventually decays back to the stable uniform state, while a perturbation that is above threshold forms a stable LS (d). (e) Initial perturbation in $c_v$ ($c_h$ inset) for the cases shown in (c) and (d), bottom light and black lines, respectively. The threshold of the perturbation (dashed line) is characterized by both width and height. The bottom/top arrows indicate the respective temporal evolution direction in (c) and (d), while the top light line depicts the asymptotic LS profile in (d). Numerical calculations throughout this paper refer to periodic domains $x\in [-50,50]$ and time intervals $t\in [0,{10}^5]$ while (c)–(e) show a zoom into the space-time behavior within the peak region showed in (b). The parameters are given in Table .Reuse & Permissions

Figure 5

(a) Profiles of localized solutions for different coupling parameter $C_p$ which are given in (b). We denote the full width of the LS by $L$, and the width of the “fronts” by $L_f$, where largest $L$ corresponds to largest $C_p$ on the top branch and vise versa. In (b) we show that the width (quantified by the overall area) increases with the coupling parameter, above a critical value $C_{p,c}$. (c) Plot of the region marked by I in (a), to show that the change in the concentration field in the profile follows an exponential behavior, which is well described by the linear analysis [Eqs. (13)].Reuse & Permissions

Figure 6

(a) Sample images extracted from the supporting video S1 [55], showing the dynamics of the intensity of fluorescently labeled myosin-IIIa near the cell membrane. These images illustrate how an extended perturbation in the protein concentration and membrane shape grows but eventually decays back to the uniform state (overall time 3 min). In (b) we plot the area ($A_{ex}$) of the high-intensity region of myosin-IIIa near the membrane, extracted from the video (supporting videos S1, S2 [55]). This measurement represents approximately the projected area of the membrane protrusion, which we compare to the calculated area under the membrane ($A_{cal}$) (c). (d)–(f) Similar to (a)–(d) for a region of the membrane where the perturbation was observed to grow and form a stable filopodia (overall time 5.2 min). Both in the experiment (e) and the theory (f) we observe an “overshoot” in the membrane area of the filopodia protrusion. Scale bars in (a) and (d) are $2\mu$m.Reuse & Permissions

Figure 7

Interaction between perturbations. Space-time plots of the calculated membrane height evolution, for the case of two local perturbations in the concentration fields, where each perturbation alone is below threshold [as in Fig. 4c]. In (a) the two perturbations interact but are initially too far from each other and eventually decay to the uniform steady state. In (c) the same two perturbations are closer, and after coalescing they evolve to the LS. In (b) and (d) we show images from a movie (supporting video S2 [55]) where myosin-IIIa is fluorescently labeled, indicating the dynamics of bundled actin at the cell edge. In (b) two initial perturbations fail to coalesce and decay, while in (d) they are close enough to coalesce and form a filopodia. The overall time shown in (b) is 5.3 min, and in (d) 2.3 min. Scale bars in (b) and (d) are $0.5\mu$m.Reuse & Permissions