First passage of molecular motors on networks of cytoskeletal filaments

Molecular motors facilitate intracellular transport through a combination of passive motion in the cytoplasm and active transport along cytoskeletal filaments. Although the motion of motors on individual filaments is often well characterized, it remains a challenge to understand their transport on networks of filaments. Here we use computer simulations of a stochastic jump process to determine first-passage times (FPTs) of a molecular motor traversing an interval containing randomly distributed filaments of fixed length. We characterize the mean first-passage time (MFPT) as a function of the number and length of filaments. Intervals containing moderate numbers of long filaments lead to the largest MFPTs with the largest relative standard deviation; in this regime, some filament configurations lead to anomalously large FPTs due to spatial regions where motors become trapped for long times. For specific filament configurations, we systematically reverse the directionality of single filaments and determine the MFPT of the perturbed configuration. Surprisingly, altering a single filament can dramatically impact the MFPT, and filaments leading to the largest changes are commonly found in different regions than the traps. We conclude by analyzing the mean square displacement of motors in unconfined systems with a large density of filaments and show that they behave diffusively at times substantially less than the MFPT to traverse the interval. However, the effective diffusion coefficient underestimates the MFPT across the bounded interval, emphasizing the importance of local configurations of filaments on first-passage properties.

Figure 1

(a) Random configuration of filaments for a system containing 100 filaments of length $2\mu \text{m}$. Filaments possess either positive (blue) or negative (green) polarity, which specifies whether a motor bound to a filament moves in the positive or negative $x$ direction. (b) Sample path of a motor traversing the system from left to right.

Figure 2

Mean first-passage time (MFPT, top) and relative standard deviation ($\sigma /\mathrm{MFPT}$, bottom) for various numbers ($N_f$) and lengths ($L_f$) of filaments. Each value is obtained from 10 000 independent trajectories, each generated with a different filament configuration.

Figure 3

Probability density of first-passage times (FPTs) in systems with no filaments (blue, MFPT = 298.9 s), 3000 filaments of length $3\mu \text{m}$ (red, MFPT = 95.2 s), and 100 filaments of length $3\mu \text{m}$ (purple, MFPT = 633.8 s). The tails of the distributions at longer times are shown in the inset. Each distribution is constructed from 10 000 trajectories.

Figure 4

Annealed versus quenched FPT distributions. The probability density is generated from independent, randomly generated network configurations (“Annealed,” MFPT = 633.8 s) and for fixed filament configurations (“Quenched 1–3” for three different configurations; MFPT = 155.7 s, 257.1 s, and 1,111.3 s, respectively). Each network contains 100 filaments of length $3\mu \text{m}$. Each distribution is constructed from 10 000 trajectories.

Figure 5

Scaled MFPT versus the fraction of negatively polarized filaments in the network. Different combinations of filament length ($L_f$) and number ($N_f$) are shown. Each curve is constructed with data from 10 000 independent trajectories (each with a randomly generated filament network). For each case, the MFPT for all trajectories is scaled to 1 (horizontal dashed line). The fraction of negative filaments is binned so that each bin contains many samples; the scaled average MFPT for each bin is shown.

Figure 6

Average residence times (a) for diffusive motion only (no filaments) and (b) for a fixed filament configuration with $N_f=100$ and $L_f=3\mu \text{m}$ (MFPT $=5484\mathrm{s}$). The heat maps show the average residence time in each square lattice site when the space is discretized; each site has area $0.01\mu {\text{m}}^2$. The corresponding line plots show the average residence time in each vertical slice of width $\mathrm{\Delta }x=0.1\mu \text{m}$. Results are averaged over 10 000 trajectories.

Figure 7

(a) The MFPT obtained when reversing the polarity of each filament (one at a time) from the configuration in Fig. 6. Results are sorted in order of increasing MFPT for filaments that were initially negatively (green) and positively (blue) polarized. The MFPT of each new configuration is averaged over 100 independent trajectories (shown with the standard deviation). The MFPT of the initial configuration is indicated by the dashed horizontal line. Welch's $t$ test is used to test the hypothesis that the MFPT associated with a perturbed filament configuration is equal to the MFPT of the original configuration. Statistically significant differences are denoted by $*$ ($p<0.05$) and $**$ ($p<0.01$). (b) Filaments colored according to their impact on the MFPT when their polarity is reversed: The fold decrease in MFPT for filaments changing from negative to positive polarity (left) and the fold increase in MFPT for filaments changing from positive to negative polarity (right).

Figure 8

Residence times and impact of single-filament reversal for five configurations of filaments with $N_f=100$ and $L_f=3\mu \text{m}$. The configurations are categorized as slow, typical, and fast in comparison to the MFPT for the annealed case. From left to right, the MFPT values for the initial configurations are 7960 s, 250 s, 251 s, 46 s, and 98 s. The first two rows show average residence times (analogous to Fig. 6). Rows 3 and 4 show the fold increase and fold decrease in MFPT, respectively, resulting from changing the polarity of single filaments [analogous to Fig. 7].

Figure 9

Mean square displacement (MSD, blue line) of motors in a $100\mu \text{m}\times 100\mu \text{m}$ domain with $20\mathrm{filaments}/\mu {\mathrm{m}}^2$. Filament lengths of $L_f=0.3\mu \mathrm{m}$ (left) and $L_f=2\mu \mathrm{m}$ (right) are shown. Dashed lines are power-law fits to long-time ($10\text{s}<t<80\text{s}$) data obtained from 1000 trajectories of length $100\mathrm{s}$. The exponents are 0.99 (left) and 1.03 (right), indicating approximately diffusive motion.

Figure 10

(a) Effective diffusion coefficient ($D_e$) obtained from the MSD for various filament lengths ($L_f$) for a system with $20\mathrm{filaments}/\mu {\mathrm{m}}^2$. The diffusion coefficient of the motor in the cytoplasm is indicated by the dashed line. The solid line is a linear fit to the data ($L_f\ge 1\mu \text{m}$), $D_e=0.81L_f$. (b) Comparison of MFPTs obtained in original simulations with filaments and in simulations without filaments using the effective diffusion coefficient. Cases with a filament network contained 2000 filaments of various lengths. Cases without filaments used the effective diffusion coefficient from (a). All data points represent averages over 10 000 trajectories.