Dynamic model for kinesin-mediated long-range transport and its local traffic jam caused by tau proteins

In neurons, several intracellular cargoes are transported by motor proteins (kinesins) which walk on microtubules (MTs). However, kinesins can possibly unbind from the MTs before they reach their destinations. The unbound kinesins randomly diffuse in neurons until they bind to MTs. Then, they walk again along the MTs to continue their tasks. Kinesins repeat this cycle of motion until they transport their cargoes to the destinations. However, most previous models mainly focused on the motion of kinesins when they walk on MTs. Thus, a new model is required to encompass the various types of kinesin motion. We developed a comprehensive model and studied the long-range axonal transport of neurons using the model. To enhance reliability of the model, it was constructed based on multiphysics on kinesin motion (i.e., chemical kinetics, diffusion, fluid dynamics, nonlinear dynamics, and stochastic characteristics). Also, parameter values for kinesin motions are carefully obtained by comparing the model predictions and several experimental observations. The axonal transport can be degraded when a large number of binding sites on MTs are blocked by excessive tau proteins. By considering the interference between walking kinesins and tau molecules on MTs, effects of tau proteins on the axonal transport are studied. One of the meaningful predictions obtained from the model is that the velocity is not an effective metric to estimate the degradation of the transport because the decrease in velocity is not noticeable when the concentration of tau protein is not high. However, our model shows that the transport locally changes near tau molecules on MTs even when the change in the velocity is not significant. Thus, a statistical method is proposed to detect this local change effectively. The advantage of this method is that a value obtained from this method is highly sensitive to the concentration of tau protein. Another benefit of this method is that this highly sensitive value can be acquired with relatively low precision and low temporal resolution considering the time scale and length scale of the kinesin motion. This method can be used to estimate the condition of the axonal transport system.

Figure 1

Kinesin and axonal long-range transport. (a) The assembly of kinesin and cargo. (b1) The structure of a neuron composed of cell body, dendrites, axon, and synapses. (b2) Various types of kinesin motion in axonal transport. (b3) The cycle of kinesin motion.

Figure 2

Brownian motion and rebinding. (a) The Brownian motion depicted from the cross section of the axon. The dots in (b) indicate the spherical volume where the unbound kinesin heads can exist with spatially uniform probability distribution. (c) The cross section of the MT. The gray layer represents the reactive zone (volume $V_{RZ}$) of the MT. (d) A cargo with a kinesin (with free heads) in the proximity of a MT. $V_{\mathrm{K}\cap \mathrm{RZ}}$ is the intersection of the allowable space $V_k$ of the unbound kinesin and the reactive zone.

Figure 3

Binding of tau molecules to tubulins. (a) The assumed configuration of bound tau molecules. (b) Three distinct types of interaction between free tau molecules and tubulins or bound tau molecules.

Figure 4

Single molecule kinetics of tau molecules and tubulins. $P_{\varphi }$ is the probability that the tau molecule is unbound, $P_{{\mathrm{TUB}}_{1-3}+\tau }$ are the probabilities that the tau molecule binds to tubulins ${\mathrm{TUB}}_{1-3}, k_{1-3}^{+}$ are the binding rate constants, $k_{1-3}^{-}$ are the unbinding rate constants, and $\left[{\mathrm{TUB}}_{1-3}\right]$ are the concentrations of ${\mathrm{TUB}}_{1-3}$.

Figure 5

Model prediction on the binding of tau molecules to MTs. (a) The binding ratio of tau molecules to MTs is shown over $m_{\tau }$. (b) Tau molecules bound on a MT. Each image is a rectangle that represents the surface of the MT. Each group of images shows the evolution of tau molecules on the surface of a MT over $m_{\tau }$ from $m_{\tau }$ = 0.05 to 1. Tubulins with no tau molecules are depicted with black dots, and tubulins with single tau are described with dark dots (or dark red). Lighter dots (or light red) are tubulins with two or more tau molecules (tau clusters). The locations of tau clusters are different in each of the groups of images (b1)–(b4) because of the stochastic characteristics on the binding of tau molecules.

Figure 6

Obtaining of the values of the parameters. (a) The binding rate of kinesins to MTs for various concentrations of tubulins. (b) The effects of $K_{a,3}$ on the distribution of molecules on the MTs are shown. (b1) Ratios of the number of tubulins occupied by two or more tau molecules to the number of tubulins of MTs over various $K_{a,3}$. The hollow circles, stars, diamonds, and filled circles denote the ratios for $m_{\tau }=0.5$, 1, 1.5, and 2, respectively. (b2) Average distance between tau clusters. (c) The ratios of the RL in the presence of tau proteins and the RL in the absence of tau proteins. The ratio is obtained from the model for various $P_{\mathrm{ub},\mathrm{cnt}}$. (c) The changes in the RL due to the parameter $P_{\mathrm{ub},\mathrm{cnt}}$. (c1) RL ratios for $m_{\tau }=0.2$; this ratio was observed to be 0.87 in previous experiments. (c2) RL ratios for $m_{\tau }=2$, and this ratio was observed to be 0.16 in previous experiments. Therefore, the value of $P_{\mathrm{ub},\mathrm{cnt}}$ is determined as 3.2 nm/ms where model predictions match experimental observations.

Figure 7

Average velocity of cargoes. (a1)–(a6) The arrangement of MTs at the cross section of the axon. (b1), (b2), and (b3) The average velocities corresponding to $\left[\tau \right]=0$, 3, and 6 $\mu \mathrm{M}$, respectively. (c1) Velocities in axons with the same $m_{\tau }$. However, $\left[\tau \right]$ and $N_{\mathrm{MT}}$ of the axons are different. The velocity for $\left[\tau \right]=3\mu \mathrm{M}$ is presented by the gray bars. The black bars denote the velocity corresponding to $\left[\tau \right]=6\mu \mathrm{M}$. The numbers in parentheses indicate $N_{\mathrm{MT}}$. (c2) The velocities over $m_{\tau }$. Circles in the inset graph represent $m_{\tau }$ corresponding to $\left[\tau \right]=3\mu \mathrm{M}$, and asterisks denote $m_{\tau }$ corresponding to $\left[\tau \right]=6\mu \mathrm{M}$.

Figure 8

Effects of tau clusters on the motion of the cargo. (a) The cycle of the motion near a tau cluster. (b) The average spatial probability distribution (ASPD) of cargoes when the tubulins located at 10 $\mu \mathrm{m}$ along the MT axis are occupied by tau clusters. Other tubulins are free of tau proteins. (c) An ASPD of an axon with $\left[\tau \right]=0\mu \mathrm{M}$ and $N_{\mathrm{MT}}=6$. (d) An ASPD of an axon with $\left[\tau \right]=6\mu \mathrm{M}$ and $N_{\mathrm{MT}}=1$. (e) The normalized mean standard deviation (std). (f) $m_{\tau }$ using the same symbols as in (e).

Figure 9

Cargo traffic. (a) $c_{\mathrm{axon}}$ over time for $\left[\tau \right]=0$, 2, 4, and 6 $\mu \mathrm{M}$. Numbers shown in circles represent ${\dot{c}}_{\mathrm{in}}$. $N_{\mathrm{MT}}$ is 1 for all these axons. (b) The rate of $c_{\mathrm{axon}}$ between 30 and 50 s. Circles, asterisks, diamonds, and triangles denote the rates of $c_{\mathrm{axon}}$ for $\left[\tau \right]=$ 0, 2, 4, and 6 $\mu \mathrm{M}$, respectively. (c) An ASCD of an axon with ${\dot{c}}_{\mathrm{in}}=5{\mathrm{s}}^{-1}$ and $\left[\tau \right]=6\mu \mathrm{M}$. (d) The normalized mean std with the same symbols as in (b).