Active bundles of polar and bipolar filaments

Bundles of actin filaments and molecular motors of the myosin family are a common subcellular organizational motif. Typically, such bundles are under contractile stress resulting from interactions between the filaments and the motors. This holds in particular for contractile rings that appear in the late stages of cell division in animal cells and that cleave the mother into two daughter cells. It was recently shown that myosin organizes into regularly spaced clusters along rings in mammalian cells, whereas myosin clusters in fission yeast travel along the perimeter of actomyosin rings [Wollrab et al., Nat. Commun. 7, 11860 (2016)]. A mechanism based on the association of the structurally polar actin filaments into bipolar structures was shown to provide a common explanation for both observations. Here, we analyze the dynamics of this mechanism in detail. We find a rich phase diagram depending on the actomyosin interaction strength and the stability of the bipolar structures. The system can notably organize into traveling waves. Furthermore, we identify the nature of the bifurcations connecting the various patterns as parameters are changed. Finally, we report experimental patterns observed in cytokinetic rings in fission yeast and link them to solutions of our dynamic equations. Our analysis highlights the possible role played by local polarity sorting of actin filaments for the dynamics and functionality of actomyosin networks.

Figure 1

Illustration of the kinetic processes captured by the dynamic Eqs. (1)–(11). Top: assembly and disassembly of bipolar filaments from polar filaments. Middle and bottom: filament displacements induced by cross-linking molecular motors. Each plus end carries a motor. Depending on the relative orientation of the linked filaments, the displacement velocity of a filament of length $\ell$ is either $\alpha$ (parallel filaments) or $\beta$ (antiparallel filaments). In the case of interactions involving bipolar filaments, the contributions of several molecular motors can add (bottom). This leads to the values of the displacement velocity that are indicated with each interacting filament pair. Red arrows indicate the direction of motion of the filaments.

Figure 2

Dynamics of myosin clusters within cytokinetic rings in fission yeast. (a) Experimental setup: fission yeast cells are inserted in microfabricated cavities to align the rings with the confocal plane of the microscope [12]. (b) Snapshots of the density of fluorescently labeled myosin (Rlc1) along parts of different rings [15]. Higher densities correspond to warmer colours; the look-up tables, contrast, and brightness optimize the visualization of clusters. Ring diameters are 2.2 $\mu \mathrm{m}$ (1, 2), 4.3 $\mu \mathrm{m}$ (3), 1.9 $\mu \mathrm{m}$ (4), and 1.7 $\mu \mathrm{m}$ (5). Time difference between subsequent snapshots are 4 s (1, 2, 4), 24 s (3), and 8 s (5). Images were acquired at 27 ${}^{\circ }\mathrm{C}$ (1, 2, 4) and 36 ${}^{\circ }\mathrm{C}$ (3, 5). (c) Kymographs corresponding to the snapshots in (b) after a polar transformation of the ring segments to a straight line. Time difference between horizontal lines: 4 s.

Figure 3

Growth rate of modes from the linearized dynamic Eq. (14) as a function of the wave number $k$. (a) At the bifurcation, the smallest wave number is the most unstable for $\beta =0.5$ and ${\omega }_d=1.8$. (b) The mode with the smallest wave number is the most unstable for $\alpha ={\alpha }_c$, but separated by an interval of stable modes another interval of unstable modes exists for some larger values of $\alpha$ for $\beta =10$ and ${\omega }_d=1.2$. (c) The most unstable mode has a finite wave number for $\beta =20$ and ${\omega }_d=1.8$. Note that for systems of size $L$ only wave numbers $k_n=2\pi n/L$ with $n=0,1,2,...$ are permitted.

Figure 4

Critical value ${\alpha }_c$ as a function of the interaction strength $\beta$ and the dissociation rate of bipolar filaments ${\omega }_d$. The homogenous state is stable for $\alpha <{\alpha }_c$. (a) The critical value as a function of the dissociation rate ${\omega }_d$ for an interaction strength $\beta =0.5$. (b) The critical value as a function of $\beta$ and ${\omega }_d$. The solid line separates the region, where the critical eigenvalue has a vanishing imaginary part (below the line) from the region, where the imaginary part does not vanish (above the line). The dashed lines indicate the cuts presented in (a) and (d). (c) Imaginary part of the critical growth exponent as a function of $\beta$ and ${\omega }_d$. The dashed and solid lines have the same meaning as in (b). (d) The critical value as a function of $\beta$ for ${\omega }_d=0.75$. System size $L=10\ell$.

Figure 5

Bifurcation diagrams. Asymptotic states are represented by the sums of the moduli of the first Fourier modes of $c^{\pm }$ and $c^{\text{bp}}$. For the oscillatory states, the mean as well as the minimal and maximal values are presented. (a) $\beta =0.5$ and ${\omega }_d=0.5$, (b) $\beta =0.5$ and $\alpha =2.11$, (c) $\beta =10$ and ${\omega }_d=0.8$. In all cases ${\omega }_c=0.1$ and $L=10$.

Figure 6

Asymptotic stationary state for $\alpha =2.06,\beta =0.5,{\omega }_d=0.5,{\omega }_c=0.1$, and $L=10$.

Figure 7

Asymptotic oscillatory state 1 (O1) for $\alpha =2.085,\beta =0.5,{\omega }_d=0.5,{\omega }_c=0.1$, and $L=10$. (a) Kymograph of the time evolution. (b, c) Density profiles for $t=420\frac{{\ell }^2}{D}$ and $t=240\frac{{\ell }^2}{D}$, respectively.

Figure 8

Asymptotic oscillatory state 2 (O2) for $\alpha =2.09,\beta =0.5,{\omega }_d=0.5,{\omega }_c=0.1$, and $L=10$. (a) Kymograph of the time evolution. (b–d) Density profiles for $t=520\frac{{\ell }^2}{D},t=400\frac{{\ell }^2}{D}$, and $t=320\frac{{\ell }^2}{D}$, respectively.

Figure 9

Transition between O1 and O2. (a, b) Orbits of the amplitudes of the first spatial Fourier modes of O1 (a) and O2 (b) for $c^{+}$ (blue, one arrow head), $c^{-}$ (green, two arrow heads), and $c^{\text{bp}}$ (red, three arrow heads). (c, d) Orbits of the first (blue, one arrow head) and second (green, two arrow heads) spatial Fourier modes of O1 (c) and O2 (d). Dashed lines link points at equal time points. (e) Orbits of the amplitude of the first Fourier mode for $c^{+}$ for different values of $\alpha$. (f) Temporal period of O1 (blue squares) and O2 (green circles) as a function of $\alpha$. Parameter values are $\alpha =2.085$ for O1 and $\alpha =2.09$ for O2 in (a–d) as well as $\beta =0.5$, ${\omega }_d=0.5$, and $L=10$.

Figure 10

Asymptotic cluster state (C) for $\alpha =2.09$, $\beta =0.5,{\omega }_d=0.5,{\omega }_c=0.1$, and $L=10$. (a) Kymograph for the evolution of an initially homogenous distribution with a 1% random perturbation. (b, c) Density profiles for $t=400\frac{{\ell }^2}{D}$ and $t=200\frac{{\ell }^2}{D}$, respectively.

Figure 11

Asymptotic traveling wave state (T) for $\alpha =1.12,\beta =0.5,{\omega }_d=1.8,{\omega }_c=0.1$, and $L=10$. (a) Kymograph of the time evolution. (b) Density profiles for $t=80\frac{{\ell }^2}{D}$.

Figure 12

Asymptotic counterpropagating traveling clusters state (TC) for $\alpha =1.28,\beta =10,{\omega }_d=0.2,{\omega }_c=0.1$, and $L=10$. (a) Kymograph of the time evolution. (b, c) Density profiles for $t=500\frac{{\ell }^2}{D}$ and $t=400\frac{{\ell }^2}{D}$, respectively.

Figure 13

Phase diagrams for $\beta =0.5$ (a) and for $\beta =10$ (b). The different phases are identified in Table 1. There is no bifurcation between the phases E (cluster state C) and K (stationary state S). The dashed line marks the inflection point of the curve representing the states amplitude as a function of $\alpha$. Other parameter values are ${\omega }_c=0.1$ and $L=10$.