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, 18 (16), 1173-83

Stepping, Strain Gating, and an Unexpected Force-Velocity Curve for Multiple-Motor-Based Transport


Stepping, Strain Gating, and an Unexpected Force-Velocity Curve for Multiple-Motor-Based Transport

Ambarish Kunwar et al. Curr Biol.


Background: Intracellular transport via processive kinesin, dynein, and myosin molecular motors plays an important role in maintaining cell structure and function. In many cases, cargoes move distances longer than expected for single motors; there is significant evidence that this increased travel is in part due to multiple motors working together to move the cargoes. Although we understand single motors experimentally and theoretically, our understanding of multiple motors working together is less developed.

Results: We theoretically investigate how multiple kinesin motors function. Our model includes stochastic fluctuations of each motor as it proceeds through its enzymatic cycle. Motors dynamically influence each other and function in the presence of thermal noise and viscosity. We test the theory via comparison with the experimentally observed distribution of step sizes for two motors moving a cargo, and by predicting slightly subadditive stalling force for two motors relative to one. In the presence of load, our predictions for travel distances and mean velocities are different from the steady-state model: with high motor-motor coupling, we predict a form of strain-gating, where-because of the underlying motor's dynamics-the motors share load unevenly, leading to increased mean travel distance of the multiple-motor system under load. Surprisingly, we predict that in the presence of small load, two-motor cargoes move slightly slower than do single-motor cargoes. Unpublished data from G.T. Shubeita, B.C. Carter, and S.P.G. confirm this prediction in vivo.

Conclusions: When only a few motors are active, fluctuations and unequal load sharing between motors can result in significant alterations of ensemble function.


Figure 1
Figure 1. Diagram of Models A (A) and Model B (B)
The cargo (yellow circle) is attached to 4 motors (colored ‘springs’ with two small blobs at their end, indicating the motors’ heads). The motors can attach and detach to the microtubule (black line). In the case shown, while there are N=4 motors attached to the cargo, there are only n=3 engaged motors, because the green motor is not bound to the microtubule. While dynamics determine the instantaneous number of engaged motors n, depending on the simulation, there is an initial choice of N between 1 and 4 motors. N does not vary once a simulation starts. The main difference between Model A (upper panel) and B (lower panel) is that Model B includes thermal noise providing ‘kicks’ in random directions, and the viscosity of the medium. Details of each model are provided in the supplement.
Figure 2
Figure 2. Predicted motions of the center of mass of the cargo
(A) Predicted distribution of steps-wise advances of the cargo, resulting from two motors moving a cargo under low load, according to the dynamics of model A (B) Predicted distribution of steps-wise advances of the cargo, resulting from two motors moving a cargo under low load, according to the dynamics of model B. Here the cargo’s motion was simulated, according to model B, and then the resulting displacement records were analyzed exactly as for experimental data, using the step-distribution method previously investigated [11]. The distribution of detected steps (red histogram bars) are compared to previous experimental data (green bars) from [11]. (C) Predicted distribution of steps-wise advances of the cargo, resulting from a single motor moving a 0.5 μm cargo under no load, but in the presence of a viscosity ~100 that of water (viscosity of water=0.001 Pa-S), according to the dynamics of model B. (D) The same as (C), but for 2 motors. The parameter values are kon = 2 × 106 M−1•sec−1, and kcat = 105 sec−1 (see Equation 1 in supplement), K0 off=55 sec−1 and dl=1.6nm (See Supplement Equation 2.4), [ATP]=3mM and B=0.029 µM (See supplement, Equation 3), A=107 and δl=1.3nm (Supplement Equation 4) and Fo = 5 pN. These parameters except Fo are same used in Supplement Figures S1 and S2 which fit extremely well to the experimentally measured single molecule data. The overall length of each motor l is chosen to be l=110 nm with compliance k=0.32 pN/nm. The ‘on’ rate (Pa) for each motor was assumed to be 5 s−1 and the rate of detachment under a load equal to or higher than stall F0 is Pback=2 s−1. These parameter values are used throughout the manuscript unless indicated otherwise; Fo = 5 pN (used in the simulation here) is a ‘tunable’ parameter, and is chosen to match our experiments; there are no other free parameters.
Figure 3
Figure 3. Predicted distributions of stall forces for cargos moved by one and two motors
Examples of traces of the cargo’s motion as it moves in an optical trap, for simulated cargos moving according to the dynamics of Model B (A), or actual beads moved by purified kinesin motors in vitro (B). In each case, the left trace is a ‘one-motor’ stall, while the right trace is a ‘two-motor’ stall. By eye, the simulated and experimental stalls appear quite similar. Multiple such traces, both simulated and experimental, were taken and analyzed according to our standard procedure [2] to yield the distribution of stalls predicted by model B (C) or experimentally observed (D). The only free parameter for the theory was the magnitude of the one-motor stall, which was tuned to yield observed stalls at approximately 4.8 pN (to match the experimentally determined one-motor stalls). For the theory, the fits of the gaussians (C) yield peak locations of 4.82 ± 0.05 pN and 8.77± 0.07 pN. For the experiments (D) the peaks are at 4.8 ± 0.1 pN and 9.0± 0.2 pN. With the exception of Fo ( = 2.8 pN), the parameter values are same as used in Figure 2.
Figure 4
Figure 4. Force-Persistence curves for cargos
For comparison with the continuum theory[4], simulations were done for motors with a single-motor stall tuned to ~5.7 pN. All models assumed and ‘on’ rate of 5 s−1 and an identical single motor unloaded velocity. A and B: The linkage stiffness was fixed (0.32 pN/nm), but there were different total number of attached motors (N). Curves shown are the predicted mean travel distance of 1, 2, 3, and 4 motors under different degrees of opposing force, moving according to Model A (A) or Model B (B). The dotted line reflects the predicted curve from the Steady State model (SS) of Klumpp et al[4].The Monte Carlo simulations (MC) started with the initial condition of all motors initially attached to microtubule. The parameters chosen for Monte Carlo Simulations of Model A and Model B are same as in Figure 2 except F0 which was tuned to stall of ~5.7 pN; F0=6 pN and F0= 5.1 pN for models A and B, respectively. The parameter values chosen for SS model are v=0.818 µm/s, πad=5 s−1, εn=1 s−1,Fd = 3 pN and Fs = 6 pN (see supplement equations (9)–(12)). C and D: Force-Persistence curves for cargos at different values of linkage stiffness, for 2 (C) and 3 (D) motors moving according to Model A. The dotted black line is the prediction of the SS model[4], for the same parameters as Fig. 4A (chosen to achieve agreement for the one-motor curve), for 2 motors (A) and three motors (B). All parameters chosen for the MC simulation of Model A are same as in Figure 4A, except for different k values as indicated. The MC initial conditions were the same as 4A, and the steady state models’ parameter values were also as in 4A. The related plot for Model B is found in the supplement, Fig. S7. For 4A–D, the velocity of single motor for SS model was chosen to be 0.818 µm/s to match the single-motor Monte Carlo simulation.
Figure 5
Figure 5. Strain Gating explored for a cargo moved by a maximum of N=2 motors
(A) A trace of forces against which motors step, for different values of k, when the cargo experiences an applied load of 9 pN. The simulation was started with initial condition that both motors are attached randomly to the microtubule and simulation was stopped when one of the motors detached. (B) The average force that a motor attached to the cargo steps against in model A, when the cargo experiences 9 pN of externally applied load, as a function of the stiffness of the linkage connecting the motors to the cargo. Initial and final conditions are identical to Figure 5A. The related curve for Model B is in the supplement, Fig. S10. (C) The distribution of forces the motors stepped against when a load of 9 pN was applied to the cargo. The 4 panels show the distribution for steps for motors linked to the cargo by different spring constants. Initial conditions identical to Figure 5A. All parameters chosen for MC simulations in Fig. 5 are same as in Figure 4C.
Figure 6
Figure 6. Force-Velocity curves
Predicted force velocity curves from our multiple motor model for N=1, 2, 3 and 4 motors from Model A (A ) and Model B (B), assuming a linkage stiffness of 0.32 pN/nm, and an ‘on’ rate of Pa=5 s−1. The parameter values and initial conditions are same as in Figure 4A. The average velocity was calculated using the velocities calculated over a time window of 0.5 sec in the steady state. (C) The force velocity curves from the steady-state (SS) model, also assuming an ‘on’ rate of 5 s−1, a single-motor stall force of 6 pN and identical single motor velocities. The parameter values chosen for SS model are v=0.818 µm/s, πad=5 s−1, εn=1 s−1,Fd = 3 pN and Fs = 6 pN (see supplement equations (7)–(8) and (10)–(12)).
Figure 7
Figure 7. Load-induced symmetry breaking results in altered mean velocity at low loads
(A) Rate of backward displacements of the cargo, as a function of single-motor processivity (upper panel), and mean cargo velocity as a function of processivity of a single motor (lower panel) for N=2 for applied load of 2 pN. Processivity was calculated using equation (6) (supplement) for different values of parameter A. The values of parameter A used in the simulation to change the effective singlemotor processivity are 67, 87, 107, 127 and 147. Other parameter values are same as in Figure 4A. The simulation was started with the initial condition that both motors are attached randomly to the microtubule. (B) Ratio of forward to rearward motor detachment events as a function of load (upper panel) and magnitude of backward displacements (after detachment of the forward motor) as a function of load (lower panel) for N=2 motors, linked to the cargo via a 0.32 pN/nm linkage. A ratio of 1 indicates equal probability of detachment of the forward vs rearward motor; ratios higher than 1 indicate the forward motor is more likely to detach than the rearward motor. The parameter values are the same as in Figure 4A. The simulation was started with the initial condition that both motors are attached randomly to the microtubule. The forcevelocity relationship presented in Fig 6(A and B) includes this effect. Note in particular that up to an externally applied load of ~4.5 pN (~3/4 max stall for one motor), the 2-motor mean velocity is below the one motor mean velocity (see Fig 6A and B, intersection of Red and Black curves). When a motor detaches, and subsequently re-attaches, the location of re-attachment is determined randomly, but constrained to be within 110 nm of the cargo, reflecting the motor’s length. (C) Cartoon of the sequence of events resulting in small backward displacements of the cargo’s center of mass. Initially, in (1) both motors are attached to the microtubule (yellow). Because the load is low, the motors are not clustered, so the location of the cargo’s center of mass (dotted red line) is determined predominantly by the forward motor (green) which balances the externally applied load (F, indicated by blue arrow). Due to the load supported by the forward motor, its processivity is decreased relative to the backward motor, and it has a higher probability of detaching (see ratio, top panel, Fig 7B). Thus, when one of the two motors detaches, it is more likely to be the forward motor, which results in backward motion of the cargo’s center of mass (2), as the externally applied load F is now supported by the only remaining attached motor (in red). However, because the cargo is held close to the microtubule by the bound motor (in red) the detached motor (in green) now has the opportunity to rebind to the microtubule. When it does (3) the distribution of its new binding locations is determined by the cargo’s current center of mass, so that in general, the rebinding of the motor occurs behind its previous attachment location. This completes the cycle, resulting in a small backward motion (Fig 7B, lower panel) of the cargo’s center of mass. Obviously, the more frequent such cycles of detachment/backward motion (determined by a combination of load, and the single-motor processivity, Fig 7A and B), the more effect there is on mean cargo velocity.

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