Predicting the stochastic guiding of kinesin-driven microtubules in microfabricated tracks: A statistical-mechanics-based modeling approach

Directional control of microtubule shuttles via microfabricated tracks is key to the development of controlled nanoscale mass transport by kinesin motor molecules. Here we develop and test a model to quantitatively predict the stochastic behavior of microtubule guiding when they mechanically collide with the sidewalls of lithographically patterned tracks. By taking into account appropriate probability distributions of microscopic states of the microtubule system, the model allows us to theoretically analyze the roles of collision conditions and kinesin surface densities in determining how the motion of microtubule shuttles is controlled. In addition, we experimentally observe the statistics of microtubule collision events and compare our theoretical prediction with experimental data to validate our model. The model will direct the design of future hybrid nanotechnology devices that integrate nanoscale transport systems powered by kinesin-driven molecular shuttles.

Figure 1

(Color online) Illustration of microtubule guiding by a microfabricated track. (a) Schematic of a kinesin-driven microtubule colliding with the track sidewall. The drawing represents an (instantaneous) snapshot during the mechanical intercounter between microtubule and the track sidewall. The microtubule approaches the wall at an angle $\theta$ $\left(0<\theta <90°\right)$ and experiences elastic bending of its free end that is detached from the kinesin-coated surface (the $X\text{−}Y$ plane). The free end may also be bent up at an angle $\varphi$ on the track sidewall plane (the $Y\text{−}Z$ plane), where its length is given by $l$ (b) Detailed geometry of the tip of the microtubule free end that is recaptured by the nearest kinesin on the channel surface (the $X\text{−}Y$ plane) due to diffusion. The curved line representing the free end is approximated with the two straight lines AB and BC so that $l\approx AB+BC=d/\text{sin}\theta +\left[d-d/\text{tan}\theta \right]$.

Figure 2

(Color online) Canonical ensemble of microstates of a microtubule characterized by its free end configuration. The microstate energy $U$ takes continuum values determined by the tip angle $\varphi$. The partition function of such a classical system, as opposed to a quantum mechanical system, is given by integrating the Boltzmann factor with respect to $\varphi$ as $Z={\int }_0^{\pi }\text{exp}\left(-\frac{U_{\theta }^{\left(d\right)}\left({\varphi }^{\prime }\right)}{k_{B}T}\right)d{\varphi }^{\prime }$.

Figure 3

(Color online) Configurations of the microtubule free end bending (a) succeeding and (b) failing the guiding process for the specific approach angle $\theta =90°$. The free end angle is $\varphi <\beta$ in (a), where the tip will subsequently be recaptured by the nearest kinesin along the track edge. In contrast, the free end angle is $\varphi >\beta$ in (b), where the energy required for the tip to be recaptured becomes so large that it will remain detached from the kinesin-coated surface.

Figure 4

(Color online) Image sequence of the microtubule guiding assays. (a) A microtubule approaching at an angle of $\theta =82°$ is guided along the CYTOP track after colliding with the sidewall. (b) A microtubule approaching at a right angle is lifted off from the kinesin-coated glass surface starting with its leading end and eventually disappears from the field of view. Scale bar is $10\mu \text{m}$.

Figure 5

(Color online) (a) Probability of success of the microtubule guiding process with respect to the microtubule approach angle $\theta$, predicted by the statistical mechanical model for various kinesin interspace distances $\left(d=50–300\text{nm}\right)$. It is predicted that the success rate becomes nearly 100% at any approach angle for a kinesin-to-kinesin distance less than 50nm. (b) Results of the guiding assays using NKHK560cys kinesin-driven microtubules, compared to a model prediction curve assuming $EI=8\times {10}^{-24}{\text{Nm}}^2$ and $d=200\text{nm}$. The error bar of each data point represents the standard deviation for the statistical distribution of microtubule guiding events observed with sample sizes ranging from 160–320.