External forces influence the elastic coupling effects during cargo transport by molecular motors

Cellular transport is achieved by the cooperative action of molecular motors which are elastically linked to a common cargo. When the motors pull on the cargo at the same time, they experience fluctuating elastic strain forces induced by the stepping of the other motors. These elastic coupling forces can influence the motors' stepping and unbinding behavior and thereby the ability to transport cargos. Based on a generic single motor description, we introduce a framework that explains the response of two identical molecular motors to a constant external force. In particular, we relate the single motor parameters, the coupling strength and the external load force to the dynamics of the motor pair. We derive four distinct transport regimes and determine how the crossover lines between the regimes depend on the load force. Our description of the overall cargo dynamics takes into account relaxational displacements of the cargo caused by the unbinding of one motor. For large forces and weak elastic coupling these back-shifts dominate the displacements. To develop an intuitive understanding about motor cooperativity during cargo transport, we introduce a time scale for load sharing. This time scale allows us to predict how the regulation of single motor parameters influences the cooperativity. As an example, we show that up-regulating the single motor processivity enhances load sharing of the motor pair.

Figure 1

Reduced state space of a cargo transported by two motors. Each cargo state is defined by the number of bound motors that pull on the cargo; in state $\left(C_0\right)$ both motors are unbound, in state $\left(C_1\right)$ and $\left(C_2\right)$ one and two motors are bound, respectively. In the latter two states, the cargo is transported via one-motor and two-motor runs with velocity $v_1$ and $v_2$, respectively. Transitions between the states are described by the unbinding rates ${\epsilon }_1$ and ${\epsilon }_2$ and by the binding rate $\pi$.

Figure 2

Cargo under an external load $F_{\mathrm{ex}}$ linked to the filament by one motor. The projection of the anchor point of the unbound motor need not match the lattice of the filament. In such a situation, the unbound, dangling motor cannot rebind into a “force-free” relaxed state. The mismatch between the projection of the anchor point and the closest binding site at the right is denoted by $\Delta x_0$.

Figure 3

Network for two elastically coupled motors exposed to an external load force $F_{\mathrm{ex}}$. Every state is associated with a certain force configuration. During one-motor runs in $\left(C_1\right)$, the total force is sustained by the single bound motor. During two-motor runs, the two motors constantly redistribute the load force as a result of their stochastic stepping. Each step acts to stretch or relax the linkers of the motors resulting in an elastic force $F_{\mathrm{K}}=\kappa l/2=Kl$ [see Eq. (3)] contributing to the total force acting on each motor. In state $\left(j\right)$ one motor is subjected to the force $F_{\mathrm{ex}}+j{F}_{\mathrm{K}}$ and the other motor to $-j{F}_{\mathrm{K}}$. We implicitly assume that an unbound motor binds in a relaxed configuration to the filament bearing no load (the corresponding rebinding transition to (0) is not included in this figure). The forces in each state determine the stepping rates $\alpha$ and the effective unbinding rate ${\omega }_{\mathrm{off}}$.

Figure 4

A load-free cargo transported by two elastically coupled motors can display four different transport regimes. These regimes depend on the elastic coupling strength between the two motors and their single motor parameters. In the weak coupling regime, no strain force is built up, and there is almost no effect on the velocity or the binding time. In contrast, for strong elastic coupling, and for $F_{\mathrm{d}}\simeq F_{\mathrm{s}}$ the force induced by the motors' stepping influences both the velocity and the binding time. However, if $F_s$ is smaller compared to $F_d$, one motor will stall the other motor before one of the motors unbinds and the system exhibits the reduced velocity regime. Finally, if $F_d$ is smaller compared to $F_s$, one motor will pull the other motor from the filament before the motors reduce their velocity, a behavior that characterizes the enhanced unbinding regime.

Figure 5

(a–d) Binding time $t_2$ of two-motor runs in units of the single motor force-free binding time $t$. (e–h) Velocity $v_2$ of two-motor runs in units of the single motor force-free velocity $v$. Both quantities are displayed as a function of the scaled detachment force $f_{\mathrm{d}}=F_{\mathrm{d}}/F_{\mathrm{K}}$ and the scaled stall force $f_{\mathrm{s}}=F_{\mathrm{s}}/F_{\mathrm{K}}$ for different scaled load forces $f_{\mathrm{ex}}=F_{\mathrm{ex}}/F_{\mathrm{K}}$; (a, e) $f_{\mathrm{ex}}=0$, (b, f) $f_{\mathrm{ex}}=3$, (c, g) $f_{\mathrm{ex}}=6$, (d, h) $f_{\mathrm{ex}}=9$. The solid black and blue lines are crossover lines between different transport regimes. The straight line corresponds to a pair of kinesin motors with a stall to detachment force ratio $F_{\mathrm{s}}/F_{\mathrm{d}}\simeq 2$. Along the latter line, the coupling parameter $K$ and the force $F_{\mathrm{ex}}$ are changed in such a way that $f_{\mathrm{ex}}=F_{\mathrm{ex}}/lK$ remains constant. The different colored segments indicate the different transport regimes (from left to right): strong coupling (red), enhanced unbinding (green), and weak coupling (blue).

Figure 6

Network to estimate two different time scales: (i) the time scale for induced unbinding, with $N_{\mathrm{p}}=\lceil f_{\mathrm{d}}-f_{\mathrm{ex}}\rceil$ and $N_{\mathrm{m}}=-\lceil f_{\mathrm{d}}\rceil$ and (ii) the time scale for reduced stepping with $N_{\mathrm{p}}=\lceil f_{\mathrm{s}}-f_{\mathrm{ex}}\rceil$ and $N_{\mathrm{m}}=-\lceil f_{\mathrm{s}}\rceil$.

Figure 7

Transport regimes for two elastically coupled motors simultaneously pulling on a cargo that is exposed to different load forces: (a) $f_{\mathrm{ex}}=0$, (b) $f_{\mathrm{ex}}=3$, (c) $f_{\mathrm{ex}}=6$, and (d) $f_{\mathrm{ex}}=9$. The lines correspond to different pairs of motors with single motor parameters given in Table 1. The color-coded line segments represent the weak coupling (blue), the strong coupling (red), the reduced velocity (black), and the enhanced unbinding regimes (green).

Figure 8

(a) Average velocity $v_2$ of two-motor runs in units of the single motor force-free velocity $v$, (b) Average duration $t_2$ of two-motor runs in units of the single motor binding time $t$ and (c) average displacement $\langle \Delta x_2\rangle$ of two-motor runs. All quantities are given as a function of the external load force $F_{\mathrm{ex}}$ for different values of the motor stiffness $\kappa$. As explained in the text, the two-motor runs consist of single steps for large values of $F_{\mathrm{ex}}$. To describe the single motors, we used the kinesin parameters, $F_{\mathrm{s}}=6\mathrm{pN},F_{\mathrm{d}}=3\mathrm{pN},v=1\mu \mathrm{m}/\mathrm{s},{\epsilon }_{}=1{\mathrm{s}}^{-1},l=8\mathrm{nm}$; see Table 1.

Figure 9

If the leading motor unbinds, the cargo relaxes to a new equilibrium position shifted by $\langle \Delta x_{\mathrm{bs}}\rangle$. Such a back-shift contributes to the overall displacement of the cargo. A similar situation occurs when the trailing motor unbinds.

Figure 10

(a–d) Overall binding time $\langle \Delta t_{\mathrm{ca}}\rangle$, (e–h) overall average velocity $v_{\mathrm{ca}}$, and (i–l) overall run length $\langle \Delta x_{\mathrm{ca}}\rangle$ of a cargo transported by two motors as a function of the scaled detachment force $f_{\mathrm{d}}$ and stall force $f_{\mathrm{s}}$ for four different load forces $f_{\mathrm{ex}}$. The straight lines correspond to a pair of kinesin motors with a stall to detachment force ratio $F_{\mathrm{s}}/F_{\mathrm{d}}\simeq 2$. Along the line, both $K$ and $F_{\mathrm{ex}}$ are changed so that $f_{\mathrm{ex}}$ remains constant. The colored line segments indicate the different transport regimes (from left to right): strong coupling (red), enhanced unbinding (green), and weak coupling (blue). The three stars represent three different systems with fixed coupling strength and fixed external load force; see Fig. 11. When a motor unbinds, the cargo relaxes to a new position. For strong coupling and large loads, these back-shifts result in an effective backward displacement; see (k) and (l).

Figure 11

Velocity (a), binding time (b), and run length (c) as a function of the external load of a cargo pulled by two kinesin motors with parameter, $F_{\mathrm{s}}=6\mathrm{pN},F_{\mathrm{d}}=3\mathrm{pN},v=1\mu \mathrm{m}/\mathrm{s},{\epsilon }_{}=1{\mathrm{s}}^{-1},l=8\mathrm{nm},\pi =5{\mathrm{s}}^{-1}$, and the elastic coupling $\kappa =0.2\mathrm{pN}/\mathrm{nm}$.

Figure 12

Steady state probability distribution of the network shown in Fig. 3 with ${\omega }_{\mathrm{off}}\left(j\right)=0$ for all $j$. The peaks of the distributions correspond to states in which the force is shared almost equally between the motors. This example corresponds to two elastically coupled kinesins with parameters $F_{\mathrm{s}}=6\mathrm{pN},F_{\mathrm{s}}=3\mathrm{pN},\kappa =0.2\mathrm{pN}/\mathrm{nm},l=8\mathrm{nm},v=1\mu \mathrm{m}/\mathrm{s},\pi =5{\mathrm{s}}^{-1}$, and ${\epsilon }_{}=1{\mathrm{s}}^{-1}$. The states are labeled as in Fig. 3.

Figure 13

(a) Load-sharing time $t_{\mathrm{sl}}$ (black line) and unbinding time ${\epsilon }_1^{-1}\left(F_{\mathrm{ex}}\right)$ (blue and red lines) for two coupled kinesins with parameters $F_{\mathrm{s}}=6\mathrm{pN},F_{\mathrm{s}}=3\mathrm{pN},\kappa =0.2\mathrm{pN}/\mathrm{nm},l=8\mathrm{nm},v=1\mu \mathrm{m}/\mathrm{s},\pi =5{\mathrm{s}}^{-1}$, and ${\epsilon }_{}=1{\mathrm{s}}^{-1}$. For smaller force-free unbinding rates, the motors have more time to share the load and thus cooperate at higher load forces. This increased cooperativity leads to higher velocities for large forces; see (b). (b) Force velocity relation for a cargo transported by two kinesin motors with two different unbinding rates: ${\epsilon }_{}=1{\mathrm{s}}^{-1}$ (red), ${\epsilon }_{}=0.3{\mathrm{s}}^{-1}$ (blue).