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, 95 (6), 2681-91

Regular Gaits and Optimal Velocities for Motor Proteins

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Regular Gaits and Optimal Velocities for Motor Proteins

R E Lee DeVille et al. Biophys J.

Abstract

It has been observed in numerical experiments that adding a cargo to a motor protein can regularize its gait. Here we explain these results via asymptotic analysis on a general stochastic motor protein model. This analysis permits a computation of various observables (e.g., the mean velocity) of the motor protein and shows that the presence of the cargo also makes the velocity of the motor nonmonotone in certain control parameters (e.g., ATP concentration). As an example, we consider the case of a single myosin-V protein transporting a cargo and show that, at realistic concentrations of ATP, myosin-V operates in the regime which maximizes motor velocity. Our analysis also suggests an experimental regimen which can test the efficacy of any specific motor protein model to a greater degree than was heretofore possible.

Figures

FIGURE 1
FIGURE 1
A schematic of the motor protein model we consider: if the motor progresses through steps af in order, then it has taken one forward step by the repeat length D. (We have shaded one of the motor heads so that the sequence of configurations is clear.) Note that frames c and d represent the same configuration; this state has been repeated in each column for comparison. The sequence of events in order are: switching from state S0 to state S1 (a to b), which corresponds to the trailing head detaching and the center of force moving by 13.5 nm; the cargo relaxing toward the motor (b to c and d), switching from state S1 to S0 (c and d to e), which corresponds to the loose head reattaching and the center of force moving by 22.5 nm; and the cargo again relaxing toward the motor (e to f). We note that while the ab and de transitions are reversible, the relaxation stages in the dynamics are not. Once the motor heads have gone through steps af, the entire motor-cargo complex has translated by 36 nm, and the orientation of the two heads has switched. The dynamics represented in this schematic are oversimplified; as can be seen in Fig. 2, it is not common that the motor-cargo complex makes one transition, and then waits through a relaxation period, and then makes another. It is much more common to observe the motor making many transitions back and forth between states during the cargo's relaxation toward the motor. The many back-and-forth transitions give rise to complications in the analysis and motivate the introduction of effective forces in the analysis below.
FIGURE 2
FIGURE 2
Dynamics of the motor protein model, with N = 2 and parameters corresponding to myosin-V as in Eq. 2. In frames ac we have chosen γ = 1 × 10−4 kg/s and [ATP] = 2 mM and in frame d we have chosen γ = 1 × 10−4 kg/s for various values of [ATP]. In frame a, we plot the position of the motor and cargo as a function of time, and in frame b we plot the force applied to the cargo versus time. In each case we have plotted the position and force when the motor is in state S0 with circles and in state S1 as stars. In frame c, we plot a histogram of the samples of the transit time (see text for precise definition) of the cargo over one repeat length (here 36 nm). Note the axes; this measurement actually has a very small variance. In fact, for these parameters, we have the mean of the transit time as 7.00 s, with standard deviation 0.19 s, giving a coefficient of variation of 2.73%; this is a very regular process. Finally, in frame d, we plot the mean velocity of the motor-cargo complex for various values of [ATP] and with γ held fixed at 1 × 10−4 kg/s. Here the circles represent the mean velocity and the error bars the standard deviation. As mentioned in the text, this velocity is not a monotone function of [ATP]; there is an optimal value of [ATP] where the velocity of the complex is ∼50% faster than at ATP saturation. Moreover, we see that for [ATP] sufficiently large the standard deviation of the velocity becomes quite small.
FIGURE 3
FIGURE 3
In frame a, we plot the theoretical mean force felt by the cargo (i.e., γ multiplied by velocity, calculated using Eqs. 6, 8, and 20) when γ is chosen sufficiently large for the steady-state analysis to be accurate. We see that there are two limits when [ATP] is chosen sufficiently large or small, but there is a intermediate range where the velocity is even larger. In frame b, we compare the results of simulations to the theoretical large-γ curve. We see that for any given fixed γ, the prediction is quite good for [ATP] large enough, and for fixed [ATP], the prediction gets better as γ increases. Also note that even though there is a positive limiting average force for the theoretical curve as [ATP] → 0, for a fixed finite γ, the velocity tails-off to zero if [ATP] becomes too small.
FIGURE 4
FIGURE 4
Two visualizations of the same data. Frame a is a series of plots of the effective force versus y for various choices of [ATP], and frame b is a surface plot versus y and [ATP]. This curve is single-peaked for [ATP] both very large and very small, but has a double peak for intermediate values of [ATP]. We have chosen some extreme (and unphysical) values of [ATP] for the purposes of illustration, but notice that there is a significant difference even between [ATP] = 2 mM and [ATP] = 2 μM.
FIGURE 5
FIGURE 5
These pictures represent the calculation of the effective force versus cargo position, and the steady-state location of the motor versus cargo position, for [ATP] = 2 M (a and b), [ATP] = 2 μM (c and d), and [ATP] = 2 pM (e and f). As mentioned before, the high and low values of ATP concentration chosen here are completely unphysical, but we choose extreme values to accentuate the differences. The left frames (a, c, and e) here are the force-versus-position graphs as shown in Fig. 4, and in the corresponding right frames (b, d, and f) we show a surface plot of the steady-state motor location probability μx(y) for y ∈ [0, 36]nm. Not surprisingly, we see in each case that as we increase the cargo's position y, the expected location of the motor increases as well. Also, the location of the sharp increasing gradients in the force profile corresponds to the location where the motor switches from one site to the next. As explained in the text, for low and high [ATP], the motor is only allowed to use half of the sites and must effectively jump by two steps at once; whereas, for intermediate [ATP], it can use the intermediate location as well.

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