Membrane Waves Driven by Actin and Myosin

We present a model which couples the membrane with the protrusive forces of actin polymerization and contractile forces of molecular motors, such as myosin. The actin polymerization at the membrane is activated by freely diffusing membrane proteins that have a spontaneous curvature. Molecular motors are recruited to the polymerizing actin filaments, from a constant reservoir, and produce a contractile force. All the forces and variables are treated in the linear limit. Our results show that for convex membrane proteins the myosin activity gives rise to robust transverse membrane waves, similar to those observed on different cells.

Figure 1

(a) Schematic description of our model (see text for details). (b) Strong contraction (large downward black arrow) leads to MP aggregation on the two sides (small arrows on the membrane), followed by the motors (dashed red arrows), leading to wave propagation.

Figure 2

The evolution of the dispersion relation $\omega (q)$; $A^{*}=0$ (blue), $A^{*}=0.9\times {10}^{-13}$ (green), $A^{*}=3\times {10}^{-11}$ (red), and $A^{*}=0.5\times {10}^{-7}$ (light blue). For the last value we plot the calculated linear dispersion from Eq. (5) (purple line). Note that for ${\omega }^{\prime \prime }(q)$ the right-hand scale relates to the purple and light blue graphs. All other parameters are given in Table . A wave forms at $q_{\omega }$ (vertical dashed line). The horizontal dash-dotted line indicates $-k_{\mathrm{off}}$. The dashed light blue lines give the dispersion using the branching myosin force kernel ${\mathcal{M}}_q$ ($L=0.5\mu \mathrm{m}$, $A^{*}=0.5\times {10}^{-7}/\Gamma$).

Figure 3

Normalized velocity-velocity correlation functions of the membrane, calculated from our model (dashed lines) and compared to the experimental data (solid lines) [4]; (a) as a function of time and (b) as a function of distance along the membrane. The dotted lines show the correlation functions when the surface tension is increased by a factor of 5. In the inset we plot the dispersion ${\omega }^{\prime }(q)$ (solid line, left scale) and ${\omega }^{\prime \prime }(q)$ (dashed line, right scale) used in our calculation (the parameters are given in Table ). The dotted line shows the dispersion relation for the larger surface tension.

Figure 4

One-dimensional simulation using our linearized model (without thermal noise), showing the formation of moving waves due to sudden membrane perturbation. We plot here the membrane height $h(x)$, myosin $m(x)$, and actin $n(x)$ density fluctuations as a function of position and time; $t=0\sec $ (light blue), $t=1\sec $ (blue), $t=2\sec $ (green), and $t=3\sec $ (red). Note that for the height profile at $t=0$ use the right-hand scale. In the inset we show the waves at longer times; $t=60\sec $ (blue), $t=90\sec $ (green), and $t=120\sec $ (red).