Yao Wang¹, Yu-Ying Liu¹, Jian Liang¹, Peng-Ye Wang², Ping Xie^,21College of Engineering and College of Science, China Agricultural University, Beijing 100083, China
²Key Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

Received:2020-09-04Revised:2020-10-22Accepted:2020-10-26Online:2021-01-07


Abstract
Many intracellular transports are performed by multiple molecular motors in a cooperative manner. Here, we use stochastic simulations to study the cooperative transport by multiple kinesin motors, focusing mainly on effects of the form of unbinding rate versus force and the rebinding rate of single motors on the cooperative transport. We consider two forms of the unbinding rate. One is the symmetric form with respect to the force direction, which is obtained according to Kramers theory. The other is the asymmetric form, which is obtained from the prior studies for the single kinesin motor. With the asymmetric form the simulated results of both velocity and run length of the cooperative transport by two identical motors and those by a kinesin-1 motor and a kinesin-2 motor are in quantitative agreement with the available experimental data, whereas with the symmetric form the simulated results are inconsistent with the experimental data. For the cooperative transport by a faster motor and a much slower motor, the asymmetric form can give both larger velocity and longer run length than the symmetric form, giving an explanation for why kinesin adopts the asymmetric form of the unbinding rate rather than the symmetric form. For the cooperative transport by two identical motors, while the velocity is nearly independent of the rebinding rate, the run length increases linearly with the rebinding rate. For the cooperative transport by two different motors, the increase of the rebinding rate of one motor also enhances the run length of the cooperative transport. The dynamics of transport by N (N=3, 4, 5, 6, 7 and 8) motors is also studied.
Keywords： molecular motors;collective transport;kinesin


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Cite this article
Yao Wang, Yu-Ying Liu, Jian Liang, Peng-Ye Wang, Ping Xie. Effects of rebinding rate and asymmetry in unbinding rate on cargo transport by multiple kinesin motors. Communications in Theoretical Physics, 2021, 73(1): 015603- doi:10.1088/1572-9494/abc46e

1. Introduction

In the cell the long-distance transports of cargoes, in forms of vesicles, organelles mitochondria, mRNA particles, liposomes, etc, are carried out by molecular motors such as kinesin and dynein [1–5]. Kinesin motors such as kinesin-1 and kinesin-2 move processively toward the plus end of microtubule (MT), transporting cargo from cell center to periphery, while cytoplasmic dynein motors move processively toward the minus end of MT, transporting cargo from cell periphery to center. In this work, we focus on the transport by kinesin motors.

The dynamics of the single kinesin motors has been studied intensively and extensively. For instance, using single-molecule experimental techniques, the stepping behaviors of the motors were revealed and the dependencies of the quantities such as velocity and processivity (characterized by run length) upon the force acting on the motors were determined [6–9]. Using theoretical analyses and numerical simulations, the experimental data on force dependencies of these quantities were quantitatively explained [10–16]. As cargoes are usually transported by several motors as a team, the cooperative transports by multiple motors were also widely studied [17–23]. For example, by designing the cargo bound with two motors, it was revealed that the velocity of the cooperative transport by two identical kinein-1 motors is close to that of the single motors, while the run length of the cooperative transport is about 1.7-fold larger than that of the single motors [21, 23]. The run length of the cooperative transport by two identical kinein-2 motors (with a rebinding rate being about 4-fold larger than kinesin-1) is about 3.7-fold larger than that of the single motors [18]. The run length of the cooperative transport by a kinesin-1 motor and a kinesin-2 motor is larger than that by two identical kinesin-1 motors [18]. By mixing kinesin-1 and a mutant one purposefully slowed 15-fold, the velocity of the cooperative transport was dominated by the faster wild-type (WT) kinesin-1 [19]. By elaborately designing the cargo bound with multiple motors, the velocity and run length of the cooperative transport by N (N=1, 2, 4, 7) motors were also studied [20]. Meanwhile, the cooperative transports by two or multiple motors were also widely studied both theoretically and numerically [24–37]. For example, by assuming that the motors share load equally, the cooperative transport by multiple motors was studied theoretically [37]. Using the single motors’ unbinding rate that is obtained from the experimental data the numerical studies of the cooperative transport by two identical kinesin-1 motors reproduced quantitatively the available experimental data of both the velocity and run length [35, 36].

Despite the extensive studies, many issues on the cooperative transport by multiple kinesin motors are still unclear. For example, the explanation of the experimental data is still lacking showing that the velocity of the cooperative transport by the faster kinesin motor and the much slower one was dominated by the faster one [19]. What are the magnitudes of the run length and stall force of the cooperative transport by the faster motor and the much slower one compared to those of the faster one? Although the experimental data showed that the cooperative transport by a kinein-1 motor and a kinesin-2 motor has a longer run length than the cooperative transport by two kinesin-1 motors [18], the theoretical or numerical explanation of the experimental data is still lacking. What is the magnitude of the velocity of the cooperative transport by a kinein-1 motor and a kinesin-2 motor compared to that of the cooperative transport by two kinesin-1 motors? Moreover, how does the variation of rebinding rate of the two single motors affect the cooperative transport by the two motors? Although the cooperative transport by multiple motors was studied theoretically and computationally [33, 36], it is necessary to make the theoretical or numerical studies using the relation of the unbinding rate versus force in both forward and backward directions which is obtained from the available experimental data showing the dramatic asymmetry in the two directions [9, 38]. The purpose of this work is to use stochastic simulations to study the cooperative transport by multiple kinesin motors, addressing the unclear issues, explaining the available experimental data and providing predicted results.

2. Methods

2.1. Single kinesin modeling

To study the cooperative transport by multiple kinesin motors, it is required to know the force dependences of the rate of forward stepping, rate of backward stepping, velocity and unbinding rate of the single kinesin motors. Here, we use the model proposed before [39–41] to present the expressions for the force dependences of the forward stepping rate (k_F), backward stepping rate (k_B) and velocity (v₁) of a kinesin motor, which can be written as(1)$\begin{eqnarray}{k}_{F}=\displaystyle \frac{{{r}_{0}}^{\left(1-F/{F}_{S}\right)}}{{{r}_{0}}^{\left(1-F/{F}_{S}\right)}+{k}^{(+)}/{k}^{(-)}}{k}^{(+)},\end{eqnarray}$(2)$\begin{eqnarray}{k}_{B}=\displaystyle \frac{1}{{{r}_{0}}^{\left(1-F/{F}_{S}\right)}+{k}^{(+)}/{k}^{(-)}}{k}^{(+)},\end{eqnarray}$(3)$\begin{eqnarray}{v}_{1}=\displaystyle \frac{{{r}_{0}}^{\left(1-F/{F}_{S}\right)}-1}{{{r}_{0}}^{\left(1-F/{F}_{S}\right)}+{k}^{(+)}/{k}^{(-)}}{k}^{(+)}d,\end{eqnarray}$where F is the force on the coiled-coil stalk of the motor and d=8 nm is the step size. The force F is defined to have a negative value when it is along the forward direction (the plus end of MT). In equations (1)–(3) there are four adjustable parameters r₀, F_S, k⁽⁺⁾ and k⁽⁻⁾, which can be determined by fitting to the available single-molecule data. Each of the four parameters has a clear physical meaning, with r₀ denoting the unloaded forward-to-backward stepping ratio (noting that from equations (1) and (2) the stepping ratio has the form $r={k}_{F}/{k}_{B}={{r}_{0}}^{\left(1-F/{F}_{S}\right)}$), F_S the stall force at which r=1 and v₁=0, k⁽⁺⁾ the ATPase rate of the trailing head and k⁽⁻⁾ the ATPase rate of the leading head. By adjusting r₀=802, F_S=8 pN, k⁽⁺⁾=97.5 ${{\rm{s}}}^{-1}$ and k⁽⁻⁾=3 ${{\rm{s}}}^{-1}$ (table 1) and with equation (3) the single-molecule data of Andreasson et al [9] on force dependence of velocity v₁ for WT kinesin-1 can be reproduced well (figure S1(a) is available online at stacks.iop.org/CTP/73/015603/mmedia in supplemental materials). For the mutant kinesin-1, we take its velocity ${v}_{1}^{({\rm{MT}})}=\,{v}_{1}^{({\rm{WT}})}/\alpha ,$ forward stepping rate ${k}_{F}^{({\rm{MT}})}={k}_{F}^{({\rm{WT}})}/\alpha $ and backward stepping rate ${k}_{B}^{({\rm{MT}})}={k}_{B}^{({\rm{WT}})}/\alpha ,$ where $\alpha $ is a constant, which is usually larger than 1, and ${v}_{1}^{({\rm{WT}})}={v}_{1},$ ${k}_{F}^{({\rm{WT}})}={k}_{F}$ and ${k}_{B}^{({\rm{WT}})}={k}_{B}$ are the velocity, forward stepping rate and backward stepping rate of the WT kinesin-1, respectively, which are calculated using equations (1)–(3) and with parameter values given in table 1. The above choice of the velocity, forward stepping rate and backward stepping rate for the mutant motor, which are reduced by the same factor relative to those for the WT case, can be understood as follows. It is considered that the mutations reduce the ATPase rates of the two kinesin heads and thus the identical mutations in the two heads have the same effect on the reduction of the two ATPase rates k⁽⁺⁾ and k⁽⁻⁾. As seen from equations (1)–(3), the velocity, forward stepping rate and backward stepping rate are therefore reduced by the same factor. On the other hand, we have checked that provided that the velocity is reduced by the same factor, reducing the forward stepping rate and backward stepping rate by different factors has nearly no effect on our results. Similarly, we consider here that a different velocity of WT kinesin-2 relative to WT kinesin-1 arises mainly from the different ATPase rate. Thus, for WT kinesin-2, for simplicity of treatment, we also take its velocity ${v}_{1}^{(2)}={v}_{1}^{(1)}/\alpha ,$ forward stepping rate ${k}_{F}^{(2)}={k}_{F}^{(1)}/\alpha $ and backward stepping rate ${k}_{B}^{(2)}={k}_{B}^{(1)}/\alpha ,$ where ${v}_{1}^{(1)}\equiv {v}_{1}^{({\rm{WT}})}\,={v}_{1},$ ${k}_{F}^{(1)}={k}_{F}^{({\rm{WT}})}={k}_{F}$ and ${k}_{B}^{(1)}={k}_{B}^{({\rm{WT}})}={k}_{B}.$


Table 1.
Table 1.Parameter values for WT kinesin-1 motor.

Parameters	value
k(+)	97.5 s−1
k(−)	3 s−1
r0	802
FS	8 pN
ϵ0	0.82 s−1
Fd	4 pN
B	870.87 nm
Fd1	0.70 pN
C	71.50 nm
Fd2	20.69 pN
μ	5 s−1
K	0.3 pN nm−1

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Here, we consider two forms for the force dependence of the unbinding rate of the single kinesin motor. One form is determined by considering the process of a Brownian particle escaping from a potential well, as usually done in the literature. According to Kramers theory, the dependence of the unbinding rate upon force can be written as(4)$\begin{eqnarray}\varepsilon ={\varepsilon }_{0}\exp \left(\displaystyle \frac{\left|F\right|}{{F}_{d}}\right),\end{eqnarray}$where ${\varepsilon }_{0}$ is the unbinding rate under no external force on the motor and F_d is the unbinding force. Since the unbinding rate described by equation (4) is symmetric with respect to the force direction, it is simply called symmetric unbinding rate.

The other form is determined from the prior experimental, theoretical and computational studies. The single-molecule data on force dependence of run length for kinesin-1 showed that the run length is dramatically asymmetric with respect to the force direction [9, 38], which was well explained computationally [42] and theoretically [43]. These studies indicate that the unbinding rate is asymmetric with respect to the force direction, and thus it is simply called asymmetric unbinding rate. For convenience, we present the expressions for the force dependence of unbinding rate under forward (F<0) and backward (F>0) forces, separately. First, we focus on F<0. The single-molecule [9] and computational [42] data on the run length can be fitted well to a two-exponential function [35](5)$\begin{eqnarray}{L}_{1}=B\exp \left(\displaystyle \frac{F}{{F}_{d1}}\right)+C\exp \left(\displaystyle \frac{F}{{F}_{d2}}\right),\,{\rm{when}}\,F\lt 0,\end{eqnarray}$where B=870.87 nm, F_d1=0.70 pN, C=71.50 nm and F_d2=20.69 pN (table 1) (figure S1(b) in supplemental materials). The unbinding rate under F<0 can then be calculated by(6)$\begin{eqnarray}\varepsilon =\displaystyle \frac{{v}_{1}}{{L}_{1}},\,{\rm{when}}\,F\lt 0,\end{eqnarray}$where v₁ and L₁ are calculated by equations (3) and (5). Second, we focus on F>0. As shown in equation (4), we simply take the following form for the force dependence of unbinding rate(7)$\begin{eqnarray}\varepsilon ={\varepsilon }_{0}\exp \left(\displaystyle \frac{F}{{F}_{d}}\right),\,{\rm{when}}\,F\geqslant 0.\end{eqnarray}$

By taking ${\varepsilon }_{0}$=0.82 ${{\rm{s}}}^{-1}$ and F_d =4 pN (table 1), and with ${L}_{1}={v}_{1}/\varepsilon $ and equation (7), the experimental [9] and computational [42] data on the force dependence of run length under F≥0 can be reproduced well (figure S1(b) in supplemental materials). It is noted that the available single-molecule data for kinesin-1 indicated that the unbinding rate could exhibit the characteristic of slip-catch-slip bond [44], which was explained quantitatively by later computational and theoretical studies [43, 45]. To be consistent with these studies, we also make calculations using the slip-catch-slip-bond form for the unbinding rate versus F (>0) (see section S1 in supplemental materials). We have checked that under no external force on the cargo, both the simple form of equation (7) and the slip-catch-slip-bond form of equations (S1) and (S2) (see section S1 in supplemental materials) for the unbinding rate of the single motors give nearly the same results on the dynamics of cooperative transports by multiple motors presented in this work. This is because under no external force on the cargo, the probability for the occurrence of the internal forces among the motors which are larger than that where the catch bond occurs is negligibly small. Under large backward external loads, the two forms give different results (see section S1 in supplemental materials). For the case of the external load on the cargo, the results calculated with the simple from of equation (7) are shown in the main text and the corresponding results calculated with the slip-catch-slip-bond form are shown in supplemental materials.

In this work, for simplicity, for all kinesin motors (including WT kinesin-1, mutant kinesin-1 and kinesin-2) we take the same value of unbinding rate under any force, implying that the run lengths of the mutant kinesin-1 and kinesin-2 are reduced by $\alpha $-fold relative to that of the WT kinesin-1.

2.2. Simulation methods for multiple coupled kinesin motors

Multiple kinesin motors are connected to a common cargo through a linker as a linear spring with an elastic coupling strength K. We take K=0.3 pN nm⁻¹ (table 1), as determined experimentally [46]. No interaction is present among motors. The distance between two neighboring motors in the equilibrium state (with zero internally elastic force) is taken to be l₀=48 nm, close to that designed in the experiments [21, 22]. We use Monte-Carlo algorithm to simulate the forward stepping, backward stepping, unbinding and rebinding of each motor, as done in our previous work [35] (see also section S2 in supplemental materials). All motors are bound initially to MT. Each motor can step forward and backward stochastically on MT with the step size d=8 nm, the forward stepping rate k_F calculated by equation (1) and the backward stepping rate k_B calculated by equation (2). Each motor can unbind stochastically from MT with the unbinding rate calculated by equations (4)–(7) (or equations (4)–(6), (S1) and (S2)) and can rebind stochastically to MT with a rebinding rate $\mu $ independent of force. The motor is considered to rebind to the MT-binding site that is nearest to the connecting point of its linker on the cargo along the movement direction. For the case of no external force acting on the cargo, after each stepping, unbinding or rebinding of one motor, the cargo is adjusted to a position with zero net force on the cargo. For the case of an external force F_ext acting on the cargo, F_ext is shared equally by the motors bound to MT, while the unbound motors experience no force (including both internal and external forces). After each stepping, unbinding or rebinding of one motor, the cargo is adjusted to a position with a net force of F_ext on the cargo. For example, when two motors are bound to MT and one motor (called motor A) makes i (i is an integer) net steps relative to the other one (called motor B), motor A experiences a force ${F}_{i,A}=\left({id}K+{F}_{{\rm{ext}}}\right)/2$ while motor B experiences a force ${F}_{{i},B}=\left(-{i}{d}K+{F}_{{\rm{ext}}}\right)/2.$ When only one motor is bound to MT, the motor experiences a force of F_ext. We take $\mu $=5 ${{\rm{s}}}^{-1}$ for WT and mutant kinesin-1 motors, as indicated experimentally [47] and used in the previous theoretical and computational works [37, 47–49]. For kinesin-2, we take its rebinding rate ${\mu }^{(2)}=\beta {\mu }^{(1)},$ where $\beta $ is a constant and ${\mu }^{(1)}$ = $\mu $=5 ${{\rm{s}}}^{-1}.$ It is mentioned that since the distance between two neighboring motors, l₀=48 nm, is large, our simulations show that any lagging motor on the track can hardly pass the leading ones. Moreover, as we have checked, the arranged order of different types of motors along the track does not affect our results.

3. Results and discussion

3.1. Asymmetric unbinding rate has advantages over symmetric unbinding rate in cooperative transport by two kinesin motors

In this section, we focus on the dynamics of the cooperative transport by two kinesin motors. One is the WT kinesin-1 with velocity v₁ and the other is the one (e.g. mutant kinesin-1) with velocity ${v}_{1}^{({\rm{MT}})}={v}_{1}/\alpha $ (see Methods). We consider both symmetric and asymmetric unbinding rates. In figures 1(a) and (b) we show the simulated results of v₂/v₁ and L₂/L₁ versus $\alpha ,$ respectively, for both symmetric and asymmetric unbinding rates under no external force, where v₂ and L₂ are velocity and run length of the cargo driven by the two motors, respectively, while v₁ and L₁ are velocity and run length of the single WT motor, respectively.

Figure 1.

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Figure 1.Dynamics of the cooperative transport by two different kinesin motors under no external force. One corresponds to WT kinesin-1 motor having velocity v₁ and rebinding rate $\mu ,$ and the other corresponds to mutant kinesin-1 motor having velocity ${v}_{1}/\alpha $ and rebinding rate $\mu .$ Lines are simulated results. Symbols are experimental data, with red circles from Rogers et al [21] and black squares from Xu et al [23]. (a) Velocity ratio v₂/v₁ versus $\alpha .$ The errors of the experimental data are calculated with Δ($v$₂/$v$₁) = $\left|\partial \left({v}_{2}/{v}_{1}\right)/\partial {v}_{1}\right|{\rm{\Delta }}{v}_{1}+\left|\partial \left({v}_{2}/{v}_{1}\right)/\partial {v}_{2}\right|{\rm{\Delta }}{v}_{2}.$ (b) Run-length ratio L₂/L₁ versus $\alpha .$ The errors of the experimental data are calculated with ${\rm{\Delta }}\left({L}_{2}/{L}_{1}\right)=\left|\partial \left({L}_{2}/{L}_{1}\right)/\partial {L}_{1}\right|{\rm{\Delta }}{L}_{1}$ $+\left|\partial \left({L}_{2}/{L}_{1}\right)/\partial {L}_{2}\right|{\rm{\Delta }}{L}_{2}.$ (c) Velocity distribution for asymmetric unbinding rate. Lines are Gaussian fits. Left, middle and right panels are the distribution for the cooperative transport by a WT kinesin motor and a mutant kinesin motor with its velocity slowed by 15-fold, the distribution for the single WT motor and the distribution for the single mutant motor, respectively. (d) Run-length distribution for asymmetric unbinding rate. Lines are single-exponential fits. Left, middle and right panels are the distribution for the cooperative transport by a WT kinesin motor and a mutant kinesin motor with its velocity slowed by 15-fold, the distribution for the single WT motor and the distribution for the single mutant motor, respectively.


First, it is seen that the simulated results of both v₂/v₁ and L₂/L₁ at $\alpha $=1, which corresponds to the case of two WT kinesin motors, for the asymmetric unbinding rate are in quantitative agreement the experimental data of Rogers et al [21] (red circles) and those of Xu et al [23] (black squares). By contrast, the simulated result of L₂/L₁ at $\alpha $=1 for the symmetric unbinding rate is deviated far away from the experimental data. These imply that the asymmetric unbinding rate is more reasonable than the symmetric unbinding rate to characterize the kinesin motors.

Second, for the symmetric unbinding rate, both v₂/v₁ and L₂/L₁ decrease largely with the increase of $\alpha $ and become leveling off at about $\alpha $>10, with v₂/v₁ decreasing from about 0.93 at $\alpha $=1 to about 0.38 at large $\alpha $ while L₂/L₁ decreasing from about 2.72 at $\alpha $=1 to about 0.73 at large $\alpha .$ By contrast, for the asymmetric unbinding rate, both v₂/v₁ and L₂/L₁ decrease slightly with the increase of $\alpha $ for $\alpha $$\geqslant $1 and become leveling off at about $\alpha $>4, with v₂/v₁ decreasing from about 0.99 at $\alpha $=1 to about 0.72 at large $\alpha $ while L₂/L₁ decreasing from about 1.76 at $\alpha $=1 to about 1.1 at large $\alpha .$ Moreover, at any value of $\alpha ,$ v₂/v₁ for the asymmetric unbinding rate is always larger than for the symmetric unbinding rate. Although L₂/L₁ at $\alpha $<3 for the asymmetric unbinding rate is smaller than for the symmetric unbinding rate, L₂/L₁ for the asymmetric unbinding rate at $\alpha $>3 is larger than for the symmetric unbinding rate. Specifically, for the asymmetric unbinding rate, we see that v₂ $\approx $ 0.72v₁ and L₂$\approx $1.1L₁ at $\alpha $=15, implying that the transport properties by a faster WT kinesin-1 and a mutant kinesin-1 with the velocity slowed by 15-fold are similar to those of the faster WT kinesin-1 (noting that the cargo velocity v₂$\approx $0.72v₁ is more than 10-fold larger than the velocity of v₁/15 for the slower motor while the cargo run length L₂$\approx $1.1L₁ is more than 16-fold larger than the run length of L₁/15 for the slower motor). This is consistent with the available experimental data showing that with mixtures of WT kinesin-1 and a mutant one purposefully slowed 15-fold, the transport velocity was dominated by the faster WT one [19]. Taken together, the above results indicate that for the system of the cargo transport by a fast motor and a slow motor, the system has a better performance (with a larger velocity and a longer run length) for the asymmetric unbinding rate than for the symmetric unbinding rate. This provides an explanation for why the kinesin motor adopts the asymmetric unbinding rate rather than the symmetric unbinding rate.

As it is known, for the single motor its velocity has approximately a Gaussian distribution [50, 51], with the peak value in the distribution being equal to the mean velocity of the motor. Then, an interesting issue for the cooperative transport by a faster motor and a slower motor is what the distribution of velocities appears. To this end, we statistically study the velocity distribution of the cooperative transport by a faster WT kinesin-1 and a mutant kinesin-1 with its velocity slowed by 15-fold for the asymmetric unbinding rate. The results are shown in figure 1(c), where for comparison we also show the velocity distribution of the single WT kinesin-1 and that of the single mutant one. It is seen that the velocity distribution of the cooperative transport by a faster motor and a slower motor also has nearly a Gaussian form, but with a wider width than the distribution for the single motor. As expected, the position of the peak value for the velocity distribution of the cooperative transport by a faster motor and a slower motor is much closer to that for the velocity distribution of the single faster motor. In addition, we also statistically study the run-length distribution of the cooperative transport by a faster WT kinesin-1 and a mutant kinesin-1 with its velocity slowed by 15-fold for the asymmetric unbinding rate. The results are shown in figure 1(d), where for comparison we also show the run-length distribution of the single WT kinesin-1 and that of the single mutant one. It is seen that the run-length distribution of the cooperative transport by a faster motor and a slower motor is approximately single exponential, resembling those of single motors.

As noted above, with the asymmetric unbinding rate the velocity and run length of the cargo transported by a faster motor and a slower one under no external force are close to those of the cargo transported by the single faster one. Then, it is interesting to see what the advantage of the cargo transport using the former design over using the latter design. To this end, we simulate the cooperative transport by a faster WT kinesin-1 and a mutant kinesin-1 with its velocity slowed by 15-fold for the asymmetric unbinding rate under an external force F_ext acting on the cargo. The results of v₂ and L₂ versus F_ext are shown in figure 2 (see also figure S2). Interestingly, we see that the stall force (under which v₂=0 and L₂=0) is about 14.2 pN, which is about 1.8-fold larger than that by the single motor (Fs=8 pN, see table 1). Thus, by adopting the asymmetric unbinding rate for the kinesin motors, while the cooperative transport by the faster and slower motors under no external force has the velocity and run length close to those by the single faster one, the cooperative transport can provide a stall force evidently larger than that by the single one.

Figure 2.

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Figure 2.Dynamics of the cooperative transport by two different kinesin motors under the external force. One motor corresponds to WT kinesin-1 having velocity v₁ and rebinding rate $\mu ,$ and the other corresponds to mutant kinesin-1 having velocity ${v}_{1}/\alpha $ ($\alpha $=15) and rebinding rate $\mu .$ The panel shows velocity v₂ and run length L₂ versus external force F_ext.

3.2. Effect of rebinding rate on cooperative transport by two kinesin motors

In this section, we focus on the effect of varying rebinding rate on the dynamics of the cooperative transport by two kinesin motors. The WT kinesin-1 has velocity v₁ and rebinding rate $\mu ,$ while the other kinesin (e.g. kinesin-2) has velocity ${v}_{1}^{({\rm{2}})}={v}_{1}/\alpha $ and rebinding rate ${\mu }^{(2)}=\beta \mu $ (see methods). Since the asymmetric unbinding rate is more reasonable than the symmetric unbinding rate, we consider only the asymmetric unbinding rate in this section.

First, we consider the cooperative transport by two identical motors. For this case, it is evident that although v₂ and L₂ are dependent on $\alpha ,$ ratios v₂/v₁ and L₂/L₁ are independent of $\alpha $ under no external force. In figures 3(a) and (b) we show the simulated results of v₂/v₁ and L₂/L₁ versus $\beta ,$ respectively, under no external force. It is seen that while the variation of the rebinding rate has nearly no effect on the cooperative transport velocity v₂, the run length L₂ increases linearly with the increase of the rebinding rate. The recent experimental data of Feng et al [18] showed that the rebinding rate of kinesin-2 is about 4-fold larger than that of kinesin-1, equivalent to $\beta $=4 for kinesin-2. Thus, from figure 3(b) we see that the simulated data of run length for both kinesin-1 at $\beta $=1 and kinesin-2 at $\beta $=4 are in good agreement with the experimental data [18]. To see the effect of varying rebinding rate on the dynamics of the cooperative transport under the external force F_ext, in figures 3(c) and (d) (see also figure S3) we show the simulated results of v₂ and L₂ versus F_ext for different values of $\beta $ and fixed $\alpha $=1. The results of the stall force versus $\beta $ are shown in inset of figure 3(c) (see also inset of figure S3(a)). It is seen interestingly that the stall force increases with the increase of the rebinding rate, but with the increase rate decreasing with the increase of the rebinding rate.

Figure 3.

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Figure 3.Dynamics of the cooperative transport by two identical kinesin motors with rebinding rate $\beta \mu .$ For kinesin-1 $\beta $=1, and for kinesi-2 $\beta $=4. Lines are simulated results. (a) Velocity ratio v₂/v₁ versus $\beta $ under no external force. (b) Run-length ratio L₂/L₁ versus $\beta $ under no external force. Symbols are experimental data from Feng et al [18], where K1–K1 represents the cooperative transport by two kinesin-1 motors and K1–K2 represents the cooperative transport by a kinesin-1 motor and a kinesin-2 motor. The errors of the experimental data are calculated with ${\rm{\Delta }}\left({L}_{2}/{L}_{1}\right)=\left|\partial \left({L}_{2}/{L}_{1}\right)/\partial {L}_{1}\right|$ ${\rm{\Delta }}{L}_{1}+\left|\partial \left({L}_{2}/{L}_{1}\right)/\partial {L}_{2}\right|{\rm{\Delta }}{L}_{2}.$ (c) Velocity v₂ and run length L₂ versus external force F_ext for $\beta $=1 and $\alpha $=1. Inset shows stall force versus $\beta .$ (d) Velocity v₂ and run length L₂ versus external force F_ext for $\beta $= 4 and $\alpha $=1.


The experimental data of Feng et al [18] showed that the run length of single kinesin-1 is 0.77 μm while that of single kinesin-2 is 0.65 μm, implying that the velocity of the single kinesin-1 is about 1.18-fold larger than that of the single kinesin-2 in our model (see Methods). Thus, it is interesting to study the cooperative transport by a motor with velocity v₁ and rebinding rate $\mu $ and a motor with velocity ${v}_{1}^{({\rm{2}})}={v}_{1}/\alpha $ ($\alpha $=1.18) and rebinding rate ${\mu }^{(2)}=\beta \mu .$ The simulated results of v₂/v₁ and L₂/L₁ versus $\beta $ under no external force are shown in figures 4(a) and (b), respectively. From figure 4(a) it is seen that v₂/v₁ decreases only slightly with $\beta $ and becomes leveling off at high $\beta .$ By contrast, from figure 4(b) it is seen that L₂/L₁ increases evidently with $\beta $ and also becomes leveling off at high $\beta .$ Specifically, at $\beta $=4 the ratio L₂/L₁ increases to a value that is about 90% of the maximum value at high $\beta .$ This implies that the further increase of $\beta $ from $\beta $=4 has only a very small contribution to the enhancement of the run length of the cooperative transport. These results indicate that $\beta $=4 for kinesin-2 is an optimal value to ensure a large velocity and a long run length for the cooperative transport by kinesin-1 and kinesin-2.

Figure 4.

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Figure 4.Dynamics of the cooperative transport by two different kinesin motors under no external force. One motor has velocity v₁ and rebinding rate $\mu ,$ and the other has velocity ${v}_{1}/\alpha $ ($\alpha $=1.18) and rebinding rate $\beta \mu .$ (a) Velocity ratio v₂/v₁ versus $\beta .$ (b) Run-length ratio L₂/L₁ versus $\beta .$ (c) Velocity for transport by single kinesin-1 motor (K1), velocity for cooperative transport by two kinesin-1 motors (K1–K1) and velocity for cooperative transport by a kinesin-1 motor and a kinesin-2 motor (K1–K2). Kinesin-1 has velocity v₁ and rebinding rate $\mu ,$ and kinesin-2 has velocity ${v}_{1}/\alpha $ ($\alpha $=1.18) and rebind rate $\beta \mu $ ($\beta $=4). (d) Run length for transport by single kinesin-1 motor (K1), run length for cooperative transport by two kinesin-1 motors (K1–K1) and run length for cooperative transport by a kinesin-1 motor and a kinesin-2 motor (K1–K2). For comparison, the experimental data of Feng et al [18] are also shown.


To make comparisons among the transport by a single kinesin-1 motor (abbreviated as K1), the cooperative transport by two kinesin-1 motors (abbreviated as K1–K1) and the cooperative transport by a kinesin-1 motor and a kinesin-2 motor (abbreviated as K1–K2), in figures 4(c) and (d) we show the simulated results of the velocity and run length, respectively, for the three systems under no external force, where for comparison the available experimental data [18] for the run length are also shown. From figures 4(c) and (d) it is seen that while the velocity has only a slight difference among the three systems, the run length for K1–K2 is evidently larger than that for K1–K1 that is larger than that for K1. Moreover, from figure 4(d) we see that our simulated results are in agreement with the available experimental data [18].

To see generally how variations of $\alpha $ and $\beta $ affect the cooperative transport by one motor with velocity v₁ and rebinding rate $\mu $ and the other motor with velocity ${v}_{1}^{({\rm{2}})}={v}_{1}/\alpha $ and rebinding rate ${\mu }^{(2)}=\beta \mu ,$ in figures 5(a) and (b) we show the simulated results of v₂/v₁ and L₂/L₁ versus $\beta ,$ respectively, for different values of $\alpha $ under no external force. From figure 5(a) it is seen that in general, for a large value of $\alpha $ the variation of $\beta $ has a larger effect on v₂/v₁ than for a small value of $\alpha .$ By contrast, for a large value of $\alpha $ the variation of $\beta $ has a smaller effect on L₂/L₁ than for a small value of $\alpha $ (figure 5(b)).

Figure 5.

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Figure 5.Dynamics of the cooperative transport by two different kinesin motors under no external force. One motor has velocity v₁ and rebinding rate $\mu ,$ and the other has velocity ${v}_{1}/\alpha $ and rebinding rate $\beta \mu .$ (a) Velocity ratio v₂/v₁ versus $\beta $ for different values of $\alpha .$ (b) Run-length ratio L₂/L₁ versus $\beta $ for different values of $\alpha .$

3.3. Cooperative transport by multiple kinesin motors

In this section, we focus on the dynamics of the cargo transport by multiple identical WT kinesin-1 motors, with each motor having velocity v₁ and rebinding rate $\mu .$ We consider no external force on the cargo.

In figures 6(a) and (b) we show the simulated results of v_N/v₁ and L_N/L₁ versus motor number N, respectively, for the asymmetric unbinding rate (black circles), where v_N and L_N are velocity and run length of the cargo, respectively. For comparison, the available experimental data [20] are also shown in figure 6 (green triangles). It is seen that while the simulated results of v_N/v₁ decreases only slightly with the increase of N (figure 6(a)), the simulated results of L_N/L₁ increases exponentially with the increase of N (figure 6(b)). The simulated data of v_N/v₁ are consistent with the experimental data (figure 6(a)). The simulated data of L_N/L₁ for N=2 and 4 are also in agreement with the experimental data (figure 6(b)). However, the simulated value of L_N/L₁ for N=7 is evidently larger than the experimental one. The origin of the deviation between the simulated and experimental data could be described as follows. In the experimental design, when N is small (e.g. N$\leqslant $4) the distance between two neighboring motors in the equilibrium state is large, whereas when N is large (e.g. N=7) the distance between two neighboring motors in the equilibrium state is small [20]. Thus, for the system with a large N (e.g. N=7), a trailing motor can often encounter its neighboring leading motor. When the trailing motor encounters the leading one, the trailing motor is likely to unbind from MT [52–54], which results in the reduction of the run length of the cargo. By contrast, in our simulations we take the distance between two neighboring motors in the equilibrium state is large (l₀= 48 nm) (see methods), a trailing motor can hardly encounter its neighboring leading one, and thus no interference among the motors is present. For comparison, we also show the simulated results for the symmetric unbinding rate, with the results being also shown in figure 6 (red squares). It is seen that v_N/v₁ for the symmetric unbinding rate has a large decrease for a given N>1 than for the asymmetric unbinding rate (figure 6(a)). L_N/L₁ for the symmetric unbinding rate increases much quicker than for the asymmetric unbinding rate (figure 6(b)), with the simulated data at a larger N being deviated far away from the experimental data. These further indicate that the symmetric unbinding rate is more unreasonable than the asymmetric unbinding rate to characterize the kinesin motors, as concluded in section 3.1.

Figure 6.

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Figure 6.Dynamics of the cooperative transport by N (N>1) identical kinesin motors under no external force. The motor has velocity v₁ and rebinding rate $\mu .$ Lines are simulated results. Symbols are experimental data from Derr et al [20]. (a) Velocity ratio v₂/v₁ versus N. The errors of the experimental data are calculated with ${\rm{\Delta }}\left({v}_{2}/{v}_{1}\right)=\left|\partial \left({v}_{2}/{v}_{1}\right)/\partial {v}_{1}\right|{\rm{\Delta }}{v}_{1}+\left|\partial \left({v}_{2}/{v}_{1}\right)/\partial {v}_{2}\right|{\rm{\Delta }}{v}_{2}.$ (b) Run-length ratio L₂/L₁ versus N. The errors of the experimental data are calculated with ${\rm{\Delta }}\left({L}_{2}/{L}_{1}\right)=\left|\partial \left({L}_{2}/{L}_{1}\right)/\partial {L}_{1}\right|{\rm{\Delta }}{L}_{1}+\left|\partial \left({L}_{2}/{L}_{1}\right)/\partial {L}_{2}\right|{\rm{\Delta }}{L}_{2}.$ (c) Simulated results of velocity distribution for asymmetric unbinding rate. Lines are Gaussian fits. Left and middle panels correspond to N=4 and 7, respectively, and right panel shows the half width of the Gaussian velocity distribution versus N. (d) Simulated results of run-length distribution for asymmetric unbinding rate. Lines are single-exponential fits. Left, middle and right panels correspond to N=4, 5 and 7, respectively.


Furthermore, we statistically study the velocity distribution and run-length distribution of the cooperative transport by N motors for the asymmetric unbinding rate. The results are shown in figures 6(c) and (d) and in figures S4 and S5 (see supplemental materials). It is seen that for any N, the velocity distribution has a Gaussian form and the run-length distribution approximately has a single-exponential form, which are in agreement with the experimental data [20]. More interestingly, it is seen that the half width of the Gaussian velocity distribution decreases with the increase of the motor number N (right panel of figure 6(c)).

4. Concluding remarks

We studied the cooperative transport by two kinesin motors using both asymmetric and symmetric unbinding rate of the single motor. Using asymmetric unbinding rate the simulated results of the velocity and run length of the cooperative transport by two kinein-1 motors are consistent with the available experimental data, whereas using symmetric unbinding rate the corresponding simulated results are inconsistent with the available experimental data. The studies show that for the cooperative transport by a faster motor and a much slower motor, the asymmetric unbinding rate can give a larger velocity and a longer run length than the symmetric unbinding rate, providing an explanation for why the kinesin motor adopts the asymmetric unbinding rate rather than the symmetric unbinding rate. With the asymmetric unbinding rate, while the cooperative transport by the faster and much slower motors under no external load has the velocity and run length close to those by the single faster one, the cooperative transport can provide a stall force nearly two-fold larger than that by the single faster one. This indicates that the cooperative transport by the faster and much slower motors has advantages over the transport by a single faster motor.

We studied the effect of the rebinding rate of the motor on the cooperative transport by two motors using asymmetric unbinding rate. The studies show that for the cooperative transport by two identical kinesin motors, while the velocity is nearly independent of the variation of the rebinding rate of the single motors, the run length increases linearly with the increase of the rebinding rate, and the stall force also increases with the increase of the rebinding rate but becomes saturated at high rebinding rate. For the cooperative transport by two different kinesin motors, the increase of the rebinding rate of one motor can also enhance the run length of the cooperative transport. The simulated results explain quantitatively the available experimental data on the cooperative transport by two kinesin-2 motors and that by a kinesin-1 motor and a kinesin-2 motor.

The studies on cooperative transport by N (N>2) kinesin motors using asymmetric unbinding rate show that the velocity is insensitive to the motor number N whereas the run length increases exponentially with the motor number N. More interestingly, the velocity distribution for any N has a Gaussian form, with the half width of the distribution decreasing with the increase of N. The velocity for any N is consistent with the available experimental data. The run length for a small N (e.g. N$\leqslant $4) is also in agreement with the available experimental data. However, for a large N (e.g. N=7) the simulated run length is evidently larger than the experimental one. This may be due to that in our simulations we choose the distance between two neighboring motors in the equilibrium state has a large value (48 nm) and thus the interference among motors are not needed to consider. The future studies are hoped to consider the distance between any two motors having a small value and consider the interference between two motors when one motor encounters the other one, explaining quantitatively the available experimental data for a large N (e.g. N=7).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11775301).

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