A Unified Walking Model for Dimeric Motor Proteins

Abstract

Dimeric motor proteins, kinesin-1, cytoplasmic dynein-1, and myosin-V, move stepwise along microtubules and actin filaments with a regular step size. The motors take backward as well as forward steps. The step ratio r and dwell time τ, which are the ratio of the number of backward steps to the number of forward steps and the time between consecutive steps, respectively, were observed to change with the load. To understand the movement of motor proteins, we constructed a unified and simple mathematical model to explain the load dependencies of r and of τ measured for the above three types of motors quantitatively. Our model consists of three states, and the forward and backward steps are represented by the cycles of transitions visiting different pairs of states among the three, implying that a backward step is not the reversal of a forward step. Each of r and τ is given by a simple expression containing two exponential functions. The experimental data for r and τ for dynein available in the literature are not sufficient for a quantitative analysis, which is in contrast to those for kinesin and myosin-V. We reanalyze the data to obtain r and τ of native dynein to make up the insufficient data to fit them to the model. Our model successfully describes the behavior of r and τ for all of the motors in a wide range of loads from large assisting loads to superstall loads.

Figure 1

A three-state model for dimeric motor proteins for a saturated concentration of ATP. (A) A diagram representing three states (0, 1, and 2) and all possible transitions. The latter are represented by directed lines or arrows with k symbols indicating the rate constants. The cycles formed by the transitions $k_{01}$ and $k_{10}$ and by $k_{02}$ and $k_{20}$, respectively, correspond to the forward and backward steps of the motor. The directed dashed lines with the symbols ${\overline{k}}_{ij}$$\left(i,j=\mathrm{0,1,2}\right)$ represent the reversals of the transitions $k_{ij}$, respectively. The vertical arrow with the symbol $k_{i\text{d}}$$\left(i=\mathrm{0,1,2}\right)$ indicates the detachment of the motor in state i from the track. (B) A simplified version of the three-state model obtained by neglecting the reverse transitions $\left({\overline{k}}_{ij}\right)$ and detachments $\left(k_{i\text{d}}\right)$ and assuming that $k_{10}$ and $k_{20}$ are much larger than $k_{01}$ and $k_{02}$, respectively. The symbols “F” and “B” indicate that transitions $k_{01}$ and $k_{02}$ correspond to forward and backward transitions, respectively. (C and D) Two types of scenarios for a walking pattern on a linear track of period $\ell$ corresponding to the model in (A); the reverse transitions and detachment processes are not shown. The motor primarily moves to the right in the absence of an applied load L. The state of the motor is labeled with a number (0, 1, and 2), whereas the chemical state of a head is labeled with a Greek letter (α, ${\alpha }^{\ast }$, and β). In model C, the two-head-bound state (state 1) involved in a forward step $\left(0\to 1\to 0\right)$ is different from that (state 2) in a backward step $\left(0\to 2\to 0\right)$, whereas the one-head-bound state (state 0) is common in these steps. In model D, by contrast, the one-head-bound state (state 1) involved in a forward step $\left(0\to 1\to 0\right)$ is different from that (state 2) in a backward step $\left(0\to 2\to 0\right)$, whereas the two-head-bound state (state 0) is common. The chemical states of the heads in state 2 are not given explicitly because the chemical pathways of backward steps have not been clarified well by experiment. To see this figure in color, go online.

Figure 2

Theoretical predictions for the dependencies of the (A) step ratio r, (B) average dwell time τ, and (C) velocity v on the load L given by Eqs. 12, 13, and 14, respectively. The parameters $a_1=$ 3.4 pN⁻¹, $a_2=$ 0.7 pN⁻¹, $L_1=$ 1.5 pN, and $L_2=$ 8.0 pN corresponding to myosin-V are used. The dashed and dash-dot lines in (A) represent the first and second expressions in Eq. 15, which are valid in the ranges of L shaded by yellow and green, respectively. The dotted, dashed, and long-dash lines in (B) and (C) represent the approximate expressions in Eqs. 16, 17, and 18, which are valid in the ranges of L indicated by yellow, purple and yellow, and blue shades, respectively. The inset of (C) shows an enlargement of the boxed region in the main panel.

Figure 3

Dependencies of the (A) step ratio r, (B) dwell time τ, and (C) velocity v on the load L for kinesin. The experimental data (symbols) were compared with the theory (solid lines), Eq. 12 for r and Eq. 13 for τ with the parameters listed in Table 1. The vertical dashed lines indicate the values of $L_1$, $L_{\text{stall}}$, $L_2$, and $L_{12}$ given in Table 1. The data indicated by NHY and CC are from (22) to (23), respectively. Kinesin from bovine brains was used in the experiment at ${25}^{\circ }\text{C}$ by Nishiyama et al. (22), and Drosophila kinesin was used in the experiment at ${23}^{\circ }\text{C}$ by Carter and Cross (23). The data on the dwell time from (23) were multiplied by a factor of 0.70 for the reason explained in the text. The dwell time data distinguished by the symbols “F” and “B” in parentheses are for those preceding forward and backward steps, respectively. The error bars represent the mean ± SE. To see this figure in color, go online.

Figure 4

Dependencies of the (A) step ratio r, (B) dwell time τ, and (C) velocity v on the load L for myosin-V. The experimental data (symbols) were compared with the theory (solid lines), Eq. 12 for r and Eq. 13 for τ with the parameters listed in Table 1. The vertical dashed lines indicate the values of $L_1$, $L_{\text{stall}}$, $L_2$, and $L_{12}$ given in Table 1. The data indicated by UHI, CVR, and GCR are from (24, 25, 26), respectively, and that indicated by SUH is from unpublished data by N. Sasaki, S. Uemura, H. Isihiwata, and H.H. presented at the 57^th Annual Meeting of the Japan Society for Cell Biology in 2004. The step ratios were calculated by reanalyzing the data of UHI and SUH. Myosin-V used by all of these groups was extracted from chick brains, but the temperature and ATP concentration were somewhat different: 23–25°C and 1 mM ATP (24), 20–23°C and 2 mM ATP (25), and 20–23°C and 100 μM ATP (26). The error bars represent the mean ± SE. To see this figure in color, go online.

Figure 5

Dependencies of the (A) step ratio r, (B) dwell time τ, and (C) velocity v on the load L for cytoplasmic dynein. The experimental data (symbols) for r and τ were compared with the theory (solid lines), Eqs. 12 and 13, respectively, with the parameters listed in Table 1. The experimental data for v were fit to the theory (dashed line) with a different set of parameters (see the text for details). The vertical dashed lines indicate the values of $L_1$, $L_{\text{stall}}$, $L_2$, and $L_{12}$ given in Table 1. The data indicated by TWH (reanalyzed in this work), NHG, and GCR are from (27, 28) to (44), respectively. The data on the dwell time from (28) were multiplied by a factor of 0.74 for the reason explained in the text. The dwell time data distinguished by the symbols “F” and “B” in parentheses are for those preceding forward and backward steps, respectively. The error bars represent the mean ± SE. To see this figure in color, go online.