Encounter Times of Intermittently Running Particles

TL;DR

The paper addresses how intermittent active transport and filament networks govern the timing of encounters between mobile intracellular partners. It uses a minimal run-and-tumble model in a disc and explicit filament networks to derive MFPT and MFET as functions of $D_{eff}=v\lambda/2$, $\lambda$, and network parameters. Key results show that increasing $\lambda$ enhances space exploration but induces a plateau in encounter times, with long-run particles more likely to meet near the boundary in unstructured domains, and that biased networks form traps that can accelerate encounters by reducing effective search dimensionality. The framework yields universal density scaling and offers a practical way to estimate intracellular encounter rates from observable motility and network features.

Abstract

Intracellular processes often rely on the timely encounter of mobile reaction partners, including intermittently motor-driven organelles. The underlying cytoskeletal network presents a complex landscape that both directs particle movement and introduces quenched disorder through filament organization. We investigate the mean first encounter times for pairs of intermittently processive and diffusive particles, moving in two dimensions with and without a fixed filament network. In unstructured domains, increasing particle run-length enhances exploration of the domain, but tends to slow down the encounter times compared to equivalent diffusing particles. Encounters for long-running particles occur preferentially near the periphery, contrasting with bulk encounters for the purely diffusive case. When particles are unbiased in their runs along dense filament networks, encounters are shown to be well approximated by a continuum run-and-tumble model. For biased particles, regions of convergent filament orientation can serve as traps that slow the overall spatial exploration but can allow for faster encounter rates by funneling particles into regions of reduced dimensionality. These findings provide a framework for estimating intracellular encounter kinetics, highlighting the role of key physical features such as the effective diffusivity, run times, and network architecture.

Figures (4)

• Figure 1: Search and encounter of run-and-tumble versus diffusive particles. Length and timescales are nondimensionalized such that domain size $R=1$ and velocity $v=1$. (a) Mean first passage time to a centrally located fixed target is plotted versus the effective diffusivity $D_\text{eff}$ of run-and-tumble particles (crosses) and diffusive particles (circles), for three different particle diameters ($a$). The run length $\lambda$ of run-and-tumble particles is varied to yield different values of $D_\text{eff}$. Dotted line gives exact analytical solution for diffusive encounter with a central target (Eq. \ref{['eq:MFPT']}). Dashed lines give the analytic $\lambda \rightarrow \infty$ limit (Eq. \ref{['eq:MFETlong']}). (b) Analogous plots of encounter times between two identical particles engaging in run-and-tumble (crosses) or diffusive (circles) motion. Dashed lines give analytic $\lambda \rightarrow \infty$ limit. Dotted vertical lines correspond to the approximate critical run-length where processive motion slows down encounter (Eq. \ref{['lambda_star']}). (c) Mean time to encounter the first of many particles present at density $\rho$. Data is plotted in dimensionless units, showing collapse for different particle densities (crosses: $\rho=10$, triangles: $\rho=100$). See Supplemental Material for simulation details.
• Figure 2: Distribution of encounter locations $P(r)$ for diffusing (solid) and running (dashed, $\lambda=1$) particles with diameter $a=0.02$. All length units are non-dimensionalized by domain radius $R=1$. The dotted line shows an analytic approximation for the encounter locations of running particles in the long-run limit. Inset shows spatial distributions of encounter locations for left: simulated diffusive particles, center: simulated run-and-tumble particles, right: analytic approximation for run-and-tumble particles. Simulation details provided in Supplemental Material.
• Figure 3: Encounter times for unbiased particles exhibiting different modes of motion. (a) Top: example trajectories are shown for diffusive particles (yellow) and run-and-tumble particles (orange); bottom: example realizations of explicit, random-stick (pink) and worm-like aster (blue) networks. (b) Mean first encounter time for pairs of particles corresponding to the motility modes illustrated in (a). Particles engaging in run-and-tumble motion or motion on explicit networks have short run lengths, with $\lambda = 0.1\mu$m. $D_\text{eff}$ is varied by changing $k_\text{on}$ in the range $10^{-1}-10^3 \text{s}^{-1}\mu\text{m}^{-1}$. (c) Analogous plots of MFET for particles with long run lengths, $\lambda = 4\mu$m. Error bars show standard error of the mean. The radius of the domain is $R=15\mu$m, the particle diameter is $a=0.2\mu$m, the random stick networks have filaments of length $5\mu$m and the worm-like aster has filaments of length $60\mu$m. Network density is $4.25 \mu$m$^{-1}$ in both cases, and the specific network structures used are illustrated in (a). Simulation details are provided in Supplemental Material.
• Figure 4: Encounter of biased particles on fixed network structures. (a) Example network structures for: (left) a WLC network with one MTOC and filament length 60 $\mu$m; (center) a WLC network with three MTOC and filaments extending to the domain boundary; (right) a randomly scattered WLC network with filament length 60 $\mu$m. Red dots mark the plus end and blue dots mark the minus end for each filament. (b) Steady-state particle spatial distributions on the three networks shown in (a), for particles biased towards the plus end ($b=1$) and those biased towards the minus end ($b=0$). (c) Mean first encounter time plotted against particle bias for particles running on the three specific networks shown in (a). The horizontal dashed line shows the mean first encounter time for diffusive particles with the same effective diffusivity. (d) Mean first encounter time for particles with $b=1$ on individual instances of random WLC networks with different numbers and lengths of filaments, but fixed filament density. MFET is plotted versus the trapped fraction, defined as the fraction of particles that fall within dense clusters in the steady-state distribution. (e) Steady-state particle distributions are shown for four example networks in (d), indicating how positioning of traps can alter the encounter time. Parameters used throughout are: $a=0.2\mu\text{m}, k_\text{on} = 10\mu\text{m}^{-1}\text{s}^{-1}, k_\text{off} = 2\text{s}^{-1}, R=15\mu\text{m}$ and filament length density $4.25\mu\text{m}^{-1}$.