Bistability, Oscillations, and Bidirectional Motion of Ensemble of Hydrodynamically Coupled Molecular Motors

We analyze the collective behavior of hydrodynamically coupled molecular motors. We show that the local fluxes induced by motor displacement can induce the experimentally observed bidirectional motion of cargoes and vesicles. By means of a mean-field approach we show that sustained oscillations as well as bistable collective motor motion arise even for very large collection of motors, when thermal noise is irrelevant. The analysis clarifies the physical mechanisms responsible for such dynamics by identifying the relevant coupling parameter and its dependence on the geometry of the hydrodynamic coupling as well as on system size. We quantify the phase diagram for the different phases that characterize the collective motion of hydrodynamically coupled motors and show that sustained oscillations can be reached for biologically relevant parameters, hence, demonstrating the relevance of hydrodynamic interactions in intracellular transport.

Figure 1

Average molecular motor velocity normalized by the single bound motor velocity $v_0=f_0/\gamma$ as a function of the dimensionless coupling $k$, as obtained from Eq. (8). Hydrodynamically (panel a) and rigidly (panel b) motor coupling, both characterized by $\mathrm{\Delta }{\omega }_{\mathrm{on}}=\mathrm{\Delta }{\omega }_{\mathrm{off}}=1$ and $\delta {\omega }_{\mathrm{on}}=\delta {\omega }_{\mathrm{off}}=1/2$ and $c_{\infty }=1$. $\lambda =0.015$, 0.05, 0.1, (square, circle, triangle) and $\lambda =0.017$, 0.033, 0.066 (square, circle, triangle) for panels (a) and (b), respectively.

Figure 2

(a) Average velocity (solid circles), velocity variance (open circles), (both normalized by the bound single motor velocity, $v_0=f_0/\gamma$), and sustained oscillations frequency, ${\omega }_v$ (upward triangles) and frequency of inversion in the bistable regime ${\mathrm{\Omega }}_v$ (downward triangles) (both normalized by the hopping rate peak value ${\omega }_0$) upon variation of $k$. ${\mathrm{\Omega }}_v$ has been magnified by a factor of 10 for sake of clearness. Hopping rates are characterized by $\mathrm{\Delta }{\omega }_{\mathrm{on},\mathrm{off}}=-1$, $\delta {\omega }_{\mathrm{on},\mathrm{off}}=2$. Inset: number of oscillation between subsequent velocity switches $N_{\omega }=2{\omega }_v/{\mathrm{\Omega }}_v$ in the bistable regime. (b) dimensionless time $\tau (k)=2v(k)/{\omega }_0\epsilon$ governing the stability of the sustained oscillation (see text) as a function of the dimensionless coupling $k$ for $\lambda =f_0/\overline{\omega }L\gamma =2\times {10}^{-2},3.3\times {10}^{-2},4\times {10}^{-2}$ (squares, circles, triangles). Points below the dashed lines are for motors undergoing sustained oscillation, whereas above the dashed line for motors in the bistable regime.

Figure 3

(a) Minimum system size, $\mathrm{\Lambda }$, calculated from Eq. (4) for symmetry breaking (solid lines) and for bistable onset (dashed lines) as a function of $l={\eta }_{2\mathrm{D},\text{mem}}/{\eta }_{3\mathrm{D},\text{cyt}}$ for motor pulling on membranes. The membrane-embedded tracer size $R$ is of the order of the membrane thickness $\sim 4\mathrm{nm}$, leading to $R\sim 1/2L$. Thicker lines stand for larger values of $k$: $k=50$, 70, 90 (solid lines) and $k=95$, 110 (dashed lines), respectively. (b) Values of the coupling parameters $k_1$, $k_2$ at which the two bifurcations occur as a function of the motor properties encoded in $\lambda =f_0/(\overline{\omega }L\gamma )$.