Transformation from Spots to Waves in a Model of Actin Pattern Formation

Actin networks in certain single-celled organisms exhibit a complex pattern-forming dynamics that starts with the appearance of static spots of actin on the cell cortex. Spots soon become mobile, executing persistent random walks, and eventually give rise to traveling waves of actin. Here we describe a possible physical mechanism for this distinctive set of dynamic transformations, by equipping an excitable reaction-diffusion model with a field describing the spatial orientation of its chief constituent (which we consider to be actin). The interplay of anisotropic actin growth and spatial inhibition drives a transformation at fixed parameter values from static spots to moving spots to waves.

Figure 1

Dynamic actin structures in the substrate-attached cortex of a Dictyostelium cell, as visualized by TIRF microscopy. $\mathrm{LimE}\Delta \mathrm{coil}\mathrm{\text{−}}\mathrm{GFP}$ was used to selectively label F-actin. The cell was incubated for 15 min under $5\mu \mathrm{M}$ latrunculin A to depolymerize F-actin. The first frame (0 min) is taken 16 minutes after reducing latrunculin A concentration to $1\mu \mathrm{M}$; the first stationary actin spots are visible (scale bar $10\mu \mathrm{m}$). Spots become more numerous and mobile (13 min frame; selected traces over a period of 30 sec shown in red). At 16 min a prominent spiral wave appears (see dashed outline). Bottom right: 4 frames of the counterclockwise-rotating spiral wave superimposed in different colors (red, green, blue, white; first and final images separated by 45 sec).

Figure 2

Stationary and moving model spots. (a) Cartoon of the actin fiber structure we conjecture for stationary and moving spots in vivo. (b) Homogeneous effective “free energy” density $f(u)$ from our model (with $u_0=0.05$, $k=4.5$ [12]) for values of actin polymerization rate $u_1$ below ($u_1=1$ [12]; solid line) at ($u_1=u_1^{\star }\approx 1.06$; dashed line) and above ($u_1=1.1$; dotted line) its first-order critical point; (c) nullclines of Eqs. (1, 2) under the same conditions. (d) $u$-field profiles ($1d$ cuts through a $2d$ box of size ${500}^2$) at three different times [$(t_1,t_2,t_3)=(5,10,20)\times {10}^3$]from three different simulations ($V_0=0$, 0.012, 0.06). We excite the medium at $t=0$ so that a spot nucleates, and we impose an orientation field $\mathit{\tau }=\widehat{\mathit{x}}/2$. The spot is stationary and stable for $V_0=0$; the spot moves persistently without significant structural distortion at the two larger values of $V_0$. We show at $t_3$ the $v$-field profiles also (thin lines). Parameter values: $u_0=0.05$, $u_1=1$, $k=4.5$, $ϵ=0.5$, $\delta =2.5$, $T_u=0$.

Figure 3

Model spot-to-wave transformation. (a) An initially static spot moves under the influence of a fixed orientation field $\mathit{\tau }=\widehat{\mathit{x}}/2$ ($V_0=0.24$) and deforms into a structure resembling a traveling wave (profile and image shown $1.05\times {10}^5$ time units after application of the field). (b) We quantify this distortion mechanism by plotting the radius of gyration $R_{\mathrm{g}}$ as a function of time for spots exposed to similar fixed $\mathit{\tau }$ and various values of $V_0$. (c) Time-ordered snapshots of $u(\mathbf{x})$ from a simulation with fluctuating $\mathit{\tau }$ starting from a single spot in the center of a box. The spot acquires orientation spontaneously (spot arrow labels spot-averaged $\mathit{\tau }$), executes a persistent random walk (numbers identify periodic box crossings; trajectories and arrows record spot motion), and begins to deform into a wave at $t\approx 24\times {10}^4$. (d) A similar simulation with nine initial spots mimics the transformation from static spots to moving spots to waves shown in Fig. 1.