Multimodal transport and dispersion of organelles in narrow tubular cells

Intracellular components explore the cytoplasm via active motor-driven transport in conjunction with passive diffusion. We model the motion of organelles in narrow tubular cells using analytical techniques and numerical simulations to study the efficiency of different transport modes in achieving various cellular objectives. Our model describes length and time scales over which each transport mode dominates organelle motion, along with various metrics to quantify exploration of intracellular space. For organelles that search for a specific target, we obtain the average capture time for given transport parameters and show that diffusion and active motion contribute to target capture in the biologically relevant regime. Because many organelles have been found to tether to microtubules when not engaged in active motion, we study the interplay between immobilization due to tethering and increased probability of active transport. We derive parameter-dependent conditions under which tethering enhances long-range transport and improves the target capture time. These results shed light on the optimization of intracellular transport machinery and provide experimentally testable predictions for the effects of transport regulation mechanisms such as tethering.

Figure 1

Schematic for the transition between particle states. Here D denotes diffusive particles, + denotes particles moving in the rightward direction, and $-$ denotes particles moving in the leftward direction. The arrows are labeled with transition rates between states.

Figure 2

Contribution of passive and active motion to spreading of particles at different length and time scales. (a) Mean square displacement for halting creepers. Black dash-dotted lines show scaling regimes. Vertical dashed lines indicate transition times between the regimes. Horizontal dashed lines indicate transition length scales. (b) Dimensionless range versus time for a halting creeper particle, with scaling regimes, transition times, and transition length scales indicated. The dotted curve shows the root mean square displacement for comparison. (c) Range versus time for typical parameter values for intracellular organelles, showing the increase in long-range transport with increasing run length, above a length scale indicated by the horizontal dashed line.

Figure 3

Dispersion of particles towards a uniform distribution via bidirectional transport. All length units are nondimensionalized by domain length $L$ and all time units by $L/v$. (a) Particle distribution density for different run lengths, at dimensionless time 0.3. (b) Entropy vs time for different run lengths. The horizontal dash-dotted line denotes the threshold entropy for the system to be considered well mixed. The vertical (green) dash-dotted line is at dimensionless time 0.3. (c) Time to reach a well-mixed state as a function of run length $\ell$ and rate of transition to an active state $\gamma$. Points $A$ and $B$ are drawn at corresponding transport parameters for lysosomes in monkey kidney cells and ${\mathrm{PrP}}^{\text{C}}$ vesicles in mouse axons, respectively (Table 1).

Figure 4

Target capture times for a single particle. (a) Cumulative encounter probability for different initial distances to the target $\widehat{x}$. The dotted and dashed lines denote the encounter probability for diffusive particles with diffusion coefficients $\widehat{D}$ and ${\widehat{D}}_{\mathrm{eff}}$, respectively. The dash-dotted line denotes the average time required for a particle in the active state to cover a distance of $\widehat{x}=1$. All length units are nondimensionalized by $\ell$ and all time units by $\ell /v$. (b) Time to reach $90\%$ capture probability for different run lengths, assuming a rapid starting rate $\gamma =1{\text{s}}^{-1}$ and distances appropriate for intracellular organelle transport. The dotted and dashed lines denote $t_{90\%}$ for diffusive particles with diffusion coefficients $\widehat{D}$ and ${\widehat{D}}_{\mathrm{eff}}$, respectively.

Figure 5

Average time for target capture by a population of uniformly distributed particles, with two different densities $\widehat{\rho }=10$ (left) and $\widehat{\rho }=0.3$ (right). All units are nondimensionalized by run length $\ell$ and run time $\ell /v$. Marked points show estimated parameters for two cellular systems: $A$, peroxisome transport in fungal hyphae [2], and $B$, vesicle transport in Aplysia axons [47] (see Table 1). Black lines mark the transition between diffusion-dominated and transport-dominated motion on the length scale of interparticle distance [Eq. (19)].

Figure 6

Effects of tethering on transport. (a) Schematic for the tethering model. The smaller cylinder denotes the region within which particles can tether to the microtubule or initiate active transport. The transition rates between states are indicated with the arrows. (b) Range vs time for weak ($K_{\text{eq}}=0.1$) and strong ($K_{\text{eq}}=500$) tethering. Dashed black lines show analytical approximations in the limits of no tethering and infinitely strong tethering, accurate for short to intermediate times [Eq. (24)]. The horizontal dash-dotted line indicates the transition length scale ${\widehat{L}}_{\text{crit}}$ where tethering becomes advantageous. (c) Average time for target capture by a population as a function of the starting rate $\widehat{\gamma }$ and binding strength $K_{\text{eq}}$. The solid line indicates the transition from diffusive to active transport as the dominant transport mode at different values of tethering strength. The dashed line shows the transition where strong tethering becomes advantageous for the target encounter.

Figure 7

Effects of state-dependent activity: (a) Time to reach $90\%$ capture probability $t_{90\%}$ for different run lengths and biologically relevant transport parameter values. Solid lines denote $t_{90\%}$ for particles that can capture their target only in the passive state. Dashed lines denote $t_{90\%}$ for particles that capture their target in either state. The $\circ$, $*$, and $†$ denote run lengths of 0.1, 1, and $10\mu \mathrm{m}$, respectively. Plots for $\ell =0.1\mu \mathrm{m}$ overlap. (b) Average time for target capture by a uniformly distributed population as a function of the starting rate $\widehat{\gamma }$ for different values of diffusivity $\widehat{D}$. The particle density $\widehat{\rho }=0.3$ and other parameter ranges include the observed parameters for peroxisome transport in fungal hyphae.

Figure 8

Schematic state diagram illustrating the particle states used to develop the analytical model for multimodal transport in a cylinder. Allowed transitions are labeled with arrows and the rates for the constant-rate transition processes (to and from the tethered or actively walking state) are indicated. The transitions between diffusive states occur with a time-varying rate that can be derived by evaluating the matrix components in Eq. (D2) at $k=0$.

Figure 9

Fano factor $\sigma /\langle t_w\rangle$ quantifying the variability in the time required for a particle to first begin a processive walk. The particle is assumed to start uniformly distributed within the inner radius $\alpha$. Results shown are for the parameters $\alpha =0.1$, $k_u=100$, and $\gamma ={10}^{-2}$.