Cooperativity of self-organized Brownian motors pulling on soft cargoes

We study the cooperative dynamics of Brownian motors moving along a one-dimensional track when an external load is applied to the leading motor, mimicking molecular motors pulling on membrane-bound cargoes in intracellular traffic. Due to the asymmetric loading, self-organized motor clusters form spontaneously. We model the motors with a two-state noise-driven ratchet formulation and study analytically and numerically the collective velocity-force and efficiency-force curves resulting from mutual interactions, mostly hard-core repulsion and weak (nonbinding) attraction. We analyze different parameter regimes including the limits of weak noise, mean-field behavior, rigid coupling, and large numbers of motors, for the different interactions. We present a general framework to classify and quantify cooperativity. We show that asymmetric loading leads generically to enhanced cooperativity beyond the simple superposition of the effects of individual motors. For weakly attracting interactions, the cooperativity is mostly enhanced, including highly coordinated motion of motors and complex nonmonotonic velocity-force curves, leading to self-regulated clusters. The dynamical scenario is enriched by resonances associated to commensurability of different length scales. Large clusters exhibit synchronized dynamics and bidirectional motion. Biological implications are discussed.

Figure 1

(Color online) Piecewise linear potential for the motor-track interaction. Gray—the zone where transitions are allowed: localized and instantaneous from $U_1$ to $U_2$ and delocalized after a time $\tau$ from $U_2$ to $U_1$. See more details in the text.

Figure 2

(Color online) Velocity-force curves for $N=1$ (circles) and $N=2$ (squares) with hard-core interactions. Curves for $N=1$ coincide with the MF approximation of $N=2$. For all curves $\alpha =1/2$, $\overline{a}=1/10$, and $\overline{\sigma }=7/40$. Lower pair of curves (solid lines): $\beta =12.65$; higher pair (dashed lines): $\beta =4$. Normalization values $V_1^0\left(0\right)$ and $F_s^0$ are obtained from Eqs. (A5, A8). Inset: logarithmic dependence with $\beta$ of the ratio between velocity at zero load and the velocity at $F^{\ast }=\frac{1}{3}{F}_s$ for a single motor. Each line corresponds to a different value of $\overline{a}$ (dashed line: $\overline{a}=0$). In all cases, $\alpha =1/2$.

Figure 3

(Color online) Nonmonotonic velocity-force curves with purely excluded volume repulsion with different values of $\overline{\sigma }$, for the minimal model; $\alpha =0.75$, $\beta =6.3$, and $\overline{a}=0$.

Figure 4

(Color online) Convergence of the velocity-force curves with the number of motors $N$ for hard-core interactions. For all curves, $\alpha =1/2$, $\beta =4$, $\overline{a}=1/10$, and $\overline{\sigma }=7/40$. Cooperativity is enhanced with an increasing number of motors until it saturates.

Figure 5

(Color online) Velocity-force curves showing reduced cooperativity for $N=2$ but then progressively enhanced cooperativity for increasing $N$; $\alpha =1/2$, $\beta =4$, $\overline{a}=1/10$, and $\overline{\sigma }=7/10$.

Figure 6

(Color online) Convergence to mean-field behavior due to long-range interactions. An increase in the characteristic length of the long-range potential $\Lambda$ destroys the enhanced cooperativity of the hard-core case, for small forces. See discussion of the crossover force in the text; $\alpha =2/3$, $\beta =6.68$, $\overline{a}=0$, and $\overline{\sigma }=0.2$.

Figure 7

(Color online) Cluster velocity decays to mean-field behavior for increasing noise strength (decreasing $\beta$); $\alpha =0.37$, $\overline{a}=0.1$, and $\overline{\sigma }=0.3$.

Figure 8

(Color online) Velocity-force curves for rigid coupling (rc). Hard-core (hc) curve for two motors is also shown (squares). Although the free motor velocity now depends on the number of motors, it quickly saturates; $\alpha =0.5$, $\beta =4$, $\overline{a}=0.1$, $\overline{\sigma }=0.235$, and $\overline{k}={10}^{-4}$.

Figure 9

(Color online) Velocity-force curves for weak attractive coupling (wc) and $N=2,3$ motors. Comparison with the hard-core (hc) case is also shown (squares). The horizontal dashed line indicates the maximum velocity of a single motor. The presence of an attractive force between the motors introduces a nonmonotonic behavior of the velocity-force relationship; $\alpha =0.18$, $\beta =14.1$, $\overline{a}=0.1$, and $\overline{\sigma }=0.2$.

Figure 10

(Color online) Velocity-force curve for weak interaction for $N=5$ motors (black dots) with the parameters of Fig. 9. Due to commensurability effects the cluster of five motors becomes highly compacted and can advance much faster than the four-motor cluster. Squares: velocity-force curve for a five-motor cluster bound with a rigid coupling for three values of the stiffness ($\overline{k}={10}^{-4},{10}^{-3},{10}^{-2}$ from left to right). We can clearly see that the behavior of the weak case approaches the one of rigid coupling. Also, on the rigid coupling, a smaller stiffness increases the overall cooperativity of the cluster. Inset: the full velocity-force curves for the rigid coupling (same three curves).

Figure 11

(Color online) The same velocity-force curve from Fig. 10 (squares) compared with the same curve for a different $\overline{\sigma }$’s (diamonds). A small change in the motor size completely removes the effect of the last motor. Previously the recruitment of the fifth motor created a huge increase in velocity; now its recruitment has no impact on the velocity. Notice how the $N=4$ (circles) and $N=5$ (diamonds) curves are almost the same now. In this case (diamonds) due to commensurability effects the fifth motor never attaches to the other four. The minimum distance between the fifth and the first motors is such that the asymmetric part of the ratchet is continuously breaking the cluster.

Figure 12

(Color online) Efficiency curve for hard-core interactions. The efficiency of the cluster increases with the number of motors until it starts to saturate; $\alpha =1/2$, $\beta =4$, $\overline{a}=1/10$, and $\overline{\sigma }=7/40$.

Figure 13

(Color online) Excitation rate per motor $r\left(F\right)=r_N\left(F\right)/N$ along a complete velocity-force curve trajectory with weak attraction, for the fully asymmetric case. We can see a dramatic change in the rate with the introduction of more motors. The rate approaches its minimum value (given by the straight line) for high velocities; $\alpha =0.15$, $\beta =14.1$, $\overline{a}=0$, and $\overline{\sigma }=0.2$.

Figure 14

(Color online) Excitation rate per motor for hard-core interaction and finite asymmetry. Although the rate does not approach its minimum value like in the fully asymmetric case, the strong reduction with the number of motors is still present; $\alpha =1/2$, $\beta =4$, $\overline{a}=1/10$, and $\overline{\sigma }=7/40$.

Figure 15

(Color online) Maximum efficiency curve for hard-core interaction. The number of motors is chosen to assure maximal efficiency. Vertical dashed lines mark the addition of one motor to the cluster. The efficiency curve for a single motor is also shown (black line); $\alpha =1/2$, $\beta =4$, $\overline{a}=1/10$, and $\overline{\sigma }=7/40$.

Figure 16

(Color online) Velocity deviations with respect to mean field (dashed line) as a function of the effective motor separation ${\overline{\sigma }}^{\ast }$, for $N=2$ and the different interactions: squares for rigid coupling (rc), diamonds for weak attractive coupling (wc), and circles for hard core (hc). Commensurability effects are evident at ${\overline{\sigma }}^{\ast }\sim 1$; $\alpha =0.93$, $\beta =6.3$, $\overline{a}=0.1$, and $F=1/4F_s$.

Figure 17

(Color online) Same as Fig. 16, but with different parameters. Commensurability effects are still evident at ${\overline{\sigma }}^{\ast }\sim 1$. The asymmetric behavior of the curve for the weak interaction is due to different mean lifetimes of the cluster; $\alpha =0.18$, $\beta =14.1$, $\overline{a}=0.1$, and $F=1/10F_s$.

Figure 18

(Color online) Probability density function for the relative positions of the motors $\left(\Delta x_i=x_1-x_i\right)$ inside a cluster with $N=10$ and hard-core interactions. Inset: zoom in on the last three-motor distributions; $F=N/2F_s$, $\alpha =0.18$, $\beta =14.1$, $\overline{\sigma }=0.2$, and $\overline{a}=0.1$.

Figure 19

(Color online) Force distribution along clusters of $N=8$ and 12 motors for $\overline{a}=0$ and $F=N/4F_s$. Circles are for hard-core interaction; the first six (ten) motors feel a force higher than the MF value, and the last two feel a much smaller force since they are not always attached to the cluster. Squares are for weak interaction; since the external force is strong, the cluster is stable and each motor shares almost the same load. Although not obvious from the force distribution, both clusters are highly cooperative. Other parameters: $\alpha =0.15$, $\beta =14.1$, and $\overline{\sigma }=0.2$.

Figure 20

(Color online) Force distribution for the different curves of Fig. 11 for $F/F_s=2.6$. Clear example of commensurability effects on the force distribution. A small change in the motor size (from circles to squares) creates a large qualitative change in the force distribution inside the cluster. Removing a motor (diamonds) moves the force distribution back to the previous form.

Figure 21

(Color online) Trajectory for a cluster with $N=6$ and hard-core interaction; $t^{\ast }=\ell /V_N\left(F\right)$. Inset: zoom in on the trajectory; $F=N/2F_s$, $\alpha =0.18$, $\beta =14.1$, $\overline{\sigma }=0.2$, and $\overline{a}=0.1$.

Figure 22

(Color online) Trajectory of the mean position of a cluster of $N=40$ motors in the bimodality zone. The system clearly shows two different velocities that are maintained for long periods of time. $t^{\ast }=\ell /V_N\left(F\right)$. Inset: velocity distribution function showing the two peaks.

Figure 23

(Color online) Velocity-force curve for $N=40$ showing the appearance of a second velocity peak and the sharp transition in the mean as the force is increased. $V_{+}$ and $V_{-}$ represent the mean value of the velocity associated with each peak of the distribution, and $V_N$ represents the global mean; $\alpha =1.85$, $\beta =1.9$, $\overline{a}=0.1$, and $\overline{\sigma }=0.37$.

Figure 24

(Color online) Schematic representation of the conditions needed to prove enhanced cooperativity. C1: starting with the two motors together in the bound state $U_1$, the first motor arrives to the transition zone and jumps to the weakly bound state $U_2$, and then both motors travel forward and after a time $\tau$ the first motor goes back to $U_1$ before the second one has reached the transition zone. C2: the first motor cannot interfere with the second one in the weakly bound state $F\tau <l-\sigma$.