Motor-mediated bidirectional transport along an antipolar microtubule bundle: A mathematical model

Long-distance bidirectional transport of organelles depends on the coordinated motion of various motor proteins on the cytoskeleton. Recent quantitative live cell imaging in the elongated hyphal cells of Ustilago maydis has demonstrated that long-range motility of motors and their endosomal cargo occurs on unipolar microtubules (MTs) near the extremities of the cell. These MTs are bundled into antipolar bundles within the central part of the cell. Dynein and kinesin-3 motors coordinate their activity to move early endosomes (EEs) in a bidirectional fashion where dynein drives motility towards MT minus ends and kinesin towards MT plus ends. Although this means that one can easily assign the drivers of bidirectional motion in the unipolar section, the bipolar orientations in the bundle mean that it is possible for either motor to drive motion in either direction. In this paper we use a multilane asymmetric simple exclusion process modeling approach to simulate and investigate phases of bidirectional motility in a minimal model of an antipolar MT bundle. In our model, EE cargos (particles) change direction on each MT with a turning rate $\Omega$ and there is switching between MTs in the bundle at the minus ends. At these ends, particles can hop between MTs with rate $q_1$ on passing from a unipolar to a bipolar section (the obstacle-induced switching rate) or $q_2$ on passing in the other direction (the end-induced switching rate). By a combination of numerical simulations and mean-field approximations, we investigate the distribution of particles along the MTs for different values of these parameters and of $\Theta$, the overall density of particles within this closed system. We find that even if $\Theta$ is low, the system can exhibit a variety of phases with shocks in the density profiles near plus and minus ends caused by queuing of particles. We discuss how the parameters influence the type of particle that dominates active transport in the bundle.

Figure 1

(a) Schematic diagram of the hyphal tip cell of Ustilago maydis where two MTs with plus ends at cell poles (tip and septum) form an antipolar bundle. (b) Schematic diagram of bidirectional transport on the bundle modeled as a discrete lattice. The two tracks in the lattice represent two MTs of lengths $1-x_1$ and $x_2$ with (no-flux) plus ends at two ends. Plus- (open) and minus-type (shaded) particles on the tracks move on separate lanes [open (shaded)] for particles moving forward to plus (minus) ends, respectively. Particles can change type (and thus lane) within the same track with rate $\omega$. The forward rate on each track is set to be 1, except at the junctions between the bipolar and unipolar sections. We assume that track switching occurs only at the junctions between sections and with rates $q_{1,2}$ ($q_{1,2}=0$ means that the MTs are uncoupled). (c) Assuming $x_1=1-x_2$, we analyze steady solutions on the lattice (b) in terms of the densities on the bipolar and unipolar sections as shown, with effective boundary conditions that emerge via self-organization of the system.

Figure 2

Phase diagram of the density profile in the unipolar section with general left boundary conditions ${\alpha }_{+}^u$ and ${\beta }_{-}^u$ and closed right boundary conditions, for $\Omega x_1<1/2$ (left panel) and $\Omega x_1>1/2$ (right panel), where $x_1=1-x_2=0.2$. Numerical examples of density profiles in the unipolar section for each phase are illustrated with indicated boundary conditions $({\alpha }_{+}^u,{\beta }_{+}^u)$. The top three density profiles in the middle column use $\Omega =1$, while the bottom density profile is using $\Omega =10$. In the density profiles, black dots are for plus-type particles, while gray dots are for the minus type.

Figure 3

Panels (a) and (b) show density profiles for LL-SL and LS-SL phases on the first track with $x_1=1-x_2=0.2$ and other parameters indicated. For plus-type (minus-type) particles, black (gray) lines show the density solution from mean-field approximations and squares (circles) show the averaged densities (over time interval $T=60000\text{s}$) using a Gillespie algorithm. Panels (c) and (d) show the transition between LL-SL and LS-SL phases for indicated $\Omega$ and $\Theta$. Lines are from the mean-field approximation (10), while circles are from numerical simulations; a shock is identified if ${\rho }_{N_1+1}^1>0.5$.

Figure 4

Density profile on the first track with parameters indicated shows an SS-HL phase. The markers for simulation and mean-field approximations are as in Figs. 3a and 3b.

Figure 5

On the left are density profiles on the first track for four different overall densities as indicated; dark lines are for plus-type particles, while gray lines are for minus-type particles. On the right is a plot of $\rho$ vs $\sigma$ from the corresponding left panel; the solid lines show the relation $J=\rho (1-\rho )-\sigma (1-\sigma )$, where $J$ is the constant net current. Deviations of the dots from the solid lines are probably due to finite-size or boundary-layer effects. Other parameters are $\Omega =5\text{,}{q}_1=0.8\text{,}\text{and}{q}_2=0.2$.

Figure 6

Density profiles on the first track for system sizes $N=300$ (+) and $N=1000$ ($•$). Black shows $\rho$, while gray shows $\sigma$. The shock width near $x=0.55$ for $\rho$ is shorter for the larger system size, while $\sigma$ shows a smooth connection through $\sigma =1/2$. Other parameters are $\Omega :=\omega N=5$, $q_1=0$, $q_2=0.5$, and $\Theta =1/2$.

Figure 7

Density profiles on the first track with indicated parameters showing smooth connection phases for intermediate overall densities. A CS phase in the bipolar section indicates that minus-type particles exhibit a smooth connection through density $1/2$ at a point where plus-type particles exhibit a shock, while for an SC phase the plus-type particles exhibit the smooth connection. The markers for simulation and mean-field approximations are as in Figs. 3a and 3b.

Figure 8

A variety of phases arise in the bundle for fixed $\Theta =7/16$ and $\Omega =5$ when varying the switching rates $q_{1,2}$. Each frame in the left and right panels represents densities of plus- and minus-type particles for fixed $q_1$ when varying $q_2$ by every 0.1 between 0 and 1. The fixed parameter $q_1\in \{0,0.2,0.5,0.8,1\}$ increases from the top to bottom frame in both panels.

Figure 9

Density profiles on the first track with indicated parameters show an HS-HS phase. The markers for simulation and mean-field approximations are as in Figs. 3a and 3b.

Figure 10

Fraction of plus-type particles in the bipolar section. On the left the lines are from the approximated equation (13) for $\Omega \in \{0.1,1,2,5,10\}$, with bolder lines for larger $\Omega$, while circles are for simulations, with larger circles for larger $\Omega$. Other parameters are $\Theta =1/16$ and $q_2=1$. The right panels examine the independence of this fraction on the overall density $\Theta$ and switching rate $q_2$ if sufficiently large; pluses and circles are for $\Theta =3/16$ and $1/16$, respectively, in both the top and bottom panels. Simulated data presented use $q_1=0.2$; quantitatively similar figures can be obtained for other $q_1$.

Figure 11

Fraction of plus-type particles $F_{+}^b$ (left panel) and the ratio of the average current for plus-type particles $R_{+}^b$ (right panel) in the bipolar section with varying $q_1$ (marked by $*$) or $q_2$ (marked by $+$) for $\Omega =5$ and $\Theta =7/16$. Simulated data presented use fixed $q_1=0.8$ ($q_2=0.1$) and varying $q_2$ ($q_1$); qualitatively similar figures can be obtained for other values of $q_{1,2}$.

Figure 12

Numerical current profiles (bottom) with corresponding density profiles (top) for overall densities $\Theta =3/16$ (left) and $\Theta =9/16$ (right). Other parameters are $\Omega =5\text{,}{q}_1=0.8\text{,}\text{and}{q}_2=0.4$. Black dots are for plus-type particles, while gray dots are for the minus type.