Spontaneous Oscillations of a Minimal Actomyosin System under Elastic Loading

Spontaneous mechanical oscillations occur in various types of biological systems where groups of motor molecules are elastically coupled to their environment. By using an optical trap to oppose the gliding motion of a single bead-tailed actin filament over a substrate densely coated with myosin motors, we mimicked this condition in vitro. We show that this minimal actomyosin system can oscillate spontaneously. Our finding accords quantitatively with a general theoretical framework where oscillatory instabilities emerge generically from the collective dynamics of molecular motors under load.

Figure 1

Gliding assay with elastic loading. The motor molecules develop an active force $F_a$ on the actin filament in a direction defined here as positive. The optical trap imposes an elastic restoring force $F=-k_{T}X$ in the negative direction. $k_T$ represents the trap stiffness and $X$ the position of the bead with respect to the center of the trap.

Figure 2

(a) Time course of bead position at the onset of interaction between actin and myosin (left) and bead trajectories in the horizontal plane before and after contact (right). In (b) and (c), from top to bottom, we show spontaneous oscillations of bead position (with corresponding trap forces), their spectral densities (smoothed and fitted with a Lorentzian function), relations between trap force and bead velocity, and position histograms. (b) Mean position, $+50\mathrm{nm}$; rms magnitude, 29 nm; peak-to-peak magnitude, $\sim 80\mathrm{nm}$; spectrum peak at ${\nu }_0=6.9\mathrm{Hz}$; quality factor $Q=1.3$. The trap stiffness was here $k_T=0.083\mathrm{pN}·{\mathrm{nm}}^{-1}$. (c) Mean position, 4.8 nm; rms magnitude, 3.2 nm; peak-to-peak magnitude, $\sim 8.5\mathrm{nm}$; spectrum peak at ${\nu }_0=2.2\mathrm{Hz}$; quality factor was $Q=2.3$. Here, $k_T=0.31\mathrm{pN}·{\mathrm{nm}}^{-1}$.

Figure 3

Stochastic simulations. (a) Periodic potentials $W_1(\xi )$ and $W_2(\xi )$ and detachment rate ${\omega }_{\mathrm{off}}(\xi )$. In (b) and (c), from top to bottom, we show spontaneous oscillations of track position, their spectral densities, the steady-state relations between the external force $\lambda v-F_a$ and the relative velocity $v$ between the track and the motors (black triangles) together with the system’s trajectory in this plane (red lines), and position histograms. (b) Frequency, ${\nu }_0=7.2\mathrm{Hz}$; rms magnitude, 30.4 nm; quality factor, $Q=1.98$. Parameter values: $l=6\mathrm{nm}$, ${\omega }_{\mathrm{on}}^{-1}=40\mathrm{ms}$, $a/l=0.3$, $d/l=0.1$, $\Omega /{\omega }_{\mathrm{on}}=200$, $U=4\mathrm{pN}·\mathrm{nm}$, $N=332$, $J=3.37\times {10}^{-27}{\mathrm{N}}^2\mathrm{s}$, $\lambda =3.69\mu \mathrm{N}·\mathrm{s}·{\mathrm{m}}^{-1}$, and $k_T=0.083\mathrm{pN}·{\mathrm{nm}}^{-1}$. (c) Frequency, ${\nu }_0=2.5\mathrm{Hz}$; rms magnitude, 2.47 nm; quality factor, $Q=8.5$. Same parameters as in (b), but for $N=10$, $J=4.83\times {10}^{-27}{\mathrm{N}}^2·\mathrm{s}$, $\lambda =49.6\mu \mathrm{N}·\mathrm{s}·{\mathrm{m}}^{-1}$, and $k_T=0.31\mathrm{pN}·{\mathrm{nm}}^{-1}$.