To explore the effect of particle motility on encounter kinetics, we first consider a simple minimal model for actively transported particles. Specifically, we assume that particles of diameter a engage in ‘run-and-tumble’ behavior [47, 57], wherein they move in a straight line with velocity v for an exponentially distributed run time with average τrun. The corresponding average run length is . At the end of each run, the particles instantaneously pick a new direction uniformly at random and proceed to run in that direction. Each particle moves independently, with no interactions among the particles

Following past models of intracellular transport [32, 34, 48], we assume the particles move in a two-dimensional disc of radius R, representing an adherent cell spread out on a surface. Such cells are favored for imaging studies because their very thin peripheral region ensures that long-range particle motion occurs primarily in one plane [18, 25, 58]. When motor-driven particles approach the cell boundary they can be expected to either stop (a ‘stubborn’ boundary [47]) or transfer to another nearby filament, as at any other microtubule intersection [25, 59]. The minimalist run-and-tumble model considered in this section neglects waiting times between individual runs and spatial correlations between consecutive runs. We thus assume a scattering boundary condition, with particles selecting a new direction at random upon hitting the boundary.

Particles are initiated uniformly throughout the domain and the particle trajectories are evolved forward in discrete time-steps until they hit a fixed target, defined by the particle center approaching within distance a of the domain center. This model interpolates between effectively diffusive and primarily ballistic motion, depending on whether the run length λ is much shorter or much longer than the other length scales of interest. The effective diffusivity of a run-and-tumble particle can be defined as , based on its mean squared displacement after many tumbles in the absence of confinement [60]. Particles that engage in longer runs have a higher and will spread through a large domain more rapidly. However, this does not necessarily endow the particle with the ability to quickly find small targets [47, 61].

We note that analysis of single particle tracking data for intracellular motility often involves extracting the effective scaling and prefactor of the mean squared displacement [14, 18, 62], so that provides a minimal description of particle motility regardless of the underlying mechanism of motion. We therefore rely on as the ‘control parameter’ dictating how rapidly the particles are moving.

In figure 1(a), we plot the mean time to hit the target as a function of . As expected, active particles with very short run lengths have the same MFPT to the target as purely diffusive particles with the corresponding effective diffusivity. The diffusive target search time scales inversely with , as expected. The simulated MFPT matches to the analytic solution [46] for diffusive particles starting uniformly scattered in the domain, given by:

Figure 1. Search and encounter of run-and-tumble versus diffusive particles. Length and time scales are nondimensionalized such that domain size is R = 1 and velocity is v = 1. (a) Mean first passage time for initially uniformly distributed particles to reach a centrally located fixed target plotted versus the effective diffusivity of run-and-tumble particles (crosses) and diffusive particles (circles), for three different particle diameters a. The run length λ of run-and-tumble particles is varied to yield different values of . Dotted line gives exact analytical solution for diffusive encounter with a central target (equation (1)). Dashed lines give the analytic limit (equation (3)). (b) Analogous plots of mean first encounter times between pairs of identical particles engaging in run-and-tumble (crosses) or diffusive (circles) motion. Dashed lines give analytic limit. Dotted vertical lines correspond to the approximate critical run length where processive motion slows down encounter (equation (4)). (c) Mean time to encounter the first of many particles present at concentration c. Data is plotted in dimensionless units, showing collapse for different particle concentrations (crosses: c = 10, triangles: c = 100). Statistics are obtained from n = 500 independent particle pairs in (a), (b), and n = 10000 in (c). Simulation details can be found in supplemental material.

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When the run length λ of active particles increases, the MFPT to the target reaches a plateau. This effect can be explained by considering what happens as . Once the average run length of the particles begins to approach the size of the domain, most particles encounter the domain boundary and pick a new direction before tumbling. Thus, the boundary effectively caps the run length for the run-and-tumble particles, so that more persistent motion no longer affects the search time. Dimensional analysis indicates the limiting search time can be written as (for some function f). More explicitly, we can calculate this search time by breaking up the particle trajectory into independent segments that start at the domain boundary and continue until either the target or the boundary is hit. This process can be treated as a form of ‘exploratory dynamics’ with some splitting probability for either successfully hitting the target or resetting back to the boundary. As highlighted in recent work [11, 63], the number of attempts prior to successfully hitting the target is given by the ratio of unsuccessful to successful runs: , where is the probability of each independent run hitting the target. For the limit considered here, this probability can be computed as . Each of the unsuccessful runs has average length:

and the final successful run should have a length of approximately R. The average time to hit the target in the limit can then be approximated by

As λ increases, the search time asymptotes to this long-run limit, shown as dashed lines in figure 1(a). The transition where long run lengths become inefficient for finding the target, as compared to diffusive motion with the same dispersion rate occurs when the two analytic approximations meet, at

Notably, this transition occurs at run lengths on the order of the target size (much smaller than the domain size). For example, to encounter a m organelle within a mammalian cell or radius m, the critical run length is approximately m. This implies that in order to rapidly search for targets inside the cell, such organelles gain no benefit from persistent runs that are longer than a micron or so. This is within the range of typical vesicular organelle run lengths of 0.2–10 µm [14, 28, 29, 64]. We note that in past work, asymptotic expansion of the long-run-length limit demonstrated that the MFPT actually increases towards the asymptote with increasing λ, giving rise to an optimal run length [47]. However, this increase is very shallow, so that the target search time at is a reasonable approximation of the optimal MFPT.

In addition to the search time for a fixed central target, we consider the mean first encounter time (MFET) between two identically moving particles initiated uniformly within the disc-shaped domain. Encounter is defined to occur when the particle centers approach within a distance a of each other, and there are no interactions between independent pairs of particles. As shown in figure 1(b), the MFET exhibits a similar behavior with respect to the effective diffusivity as does the search time for a stationary target. Small run lengths yield the same encounter times as for diffusive particles, while long run lengths result in a limiting asymptote for the MFET. The approximate encounter time in the long-run limit is derived in the supplemental material.

We note that the MFET and the MFPT differ by less than two-fold (supplemental figure S1), although the diffusivity of the mobile particles relative to each other is a factor of two higher when both of them are moving. This may be due to the placement of the fixed target in the most accessible region of the domain (the center), while the mobile particles spend time near the domain boundary where access to the other particle is partially blocked.

The influence of domain boundaries on encounter times between moving particles has previously been examined in the context of particles exploring network structures [56, 65]. Two diffusing particles are more likely to find each other in the the center rather than near the boundary. For a one-dimensional domain, this effect can be demonstrated analytically (see supplemental material), and simulations indicate that an analogous preference for encounters in the bulk also applies to diffusive particles in two dimensions (figure 2).

Figure 2. Distribution of encounter locations P(r) for diffusing (solid, black) and running (dashed, ) 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 with λ = 1.0, right: analytic approximation for run-and-tumble particles with long run length. Simulation details provided in supplemental material.

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Interestingly, run-and-tumble particles with long run lengths show a very different distribution of encounter locations, with encounters occurring preferentially near the periphery of the domain. This effect has not, to the best of our knowledge, been previously reported in the literature. The transition in the spatial distribution occurs gradually as the run length increases, moving from the diffusive limit towards the analytic approximation for infinitely long runs (derived in supplemental material, see dotted curve in figure 2). The preference for peripheral encounters can be conceptually rationalized by breaking up the search process into a sequence of ‘attempts’ in which one particle starts at a boundary and the other starts distributed throughout the domain. A successful attempt is more likely to terminate in encounter near a boundary, since one particle started there. An unsuccessful attempt ends when one of the particles hits the boundary, resetting the system to try again. This provides yet another manifestation of exploratory dynamics [11], where the resetting process (hitting a boundary) modulates the steady-state behavior of the system [63].

Many cellular particles need to encounter one of a population of interaction partners, rather than a specific target. Examples include autophagosomes that must find a lysosome [6] or APPL1-bearing endosomes that must meet an EEA1-marked endosome [5] to proceed towards maturation. We therefore calculate the MFET for a fixed particle to find the first of many identical moving particles, present at a spatial concentration c. This concentration rescales the effective domain size that must be searched for the encounter. We can therefore plot the non-dimensionalized MFET () versus the non-dimensionalized effective diffusivity () and target size (). With this rescaling, different concentrations collapse together (figure 1(c)) to give an approximate universal behavior that describes how rapidly run-and-tumble particles find interaction partners among a population. As with single-target encounters, increasing the run-length speeds up the encounter for short runs, but has little effect once the run-length is sufficiently long compared to the target.

Intracellular motor-driven cargos are constrained to move along cytoskeletal filaments, whose varied configurations can guide, enhance, or impair long-range transport [19, 32, 35]. Here we incorporate explicit stable network architectures as a source of quenched disorder underlying active particle motion, examining how encounter times between pairs of moving particles are affected by the network structure.

We simulate the motion of particles running on two different types of explicit networks: a ‘random stick’ network (also known as a Mikado network [66]), composed of short straight segments randomly scattered through the domain, and a worm-like-aster network consisting of semiflexible worm-like chains (WLCs) that emanate from a central point towards the cell periphery, bending and entangling when they reach the outer edge. The former can be taken to represent an actin mesh; the latter corresponds to microtubule structures observed in mammalian cells with a perinuclear MTOC. Both networks have a spatial density of (filament length per domain area), consistent with microtubule densities observed in fish melanophores [67].

Upon binding to a filament, the particles select a direction of motion, moving towards either end with equal probability. Bound particles move with speed v, and unbind from the filament with constant rate . Particles that reach a filament end remain attached at the tip until an unbinding event occurs. When the end is adjacent to a domain boundary, this is akin to a stubborn boundary condition [47].

While unbound, the particles diffuse freely with diffusivity D. They are able to rebind to nearby filaments at rate , where is the length of the filament found within contact distance from the particle center. Diffusive particles treat the domain edge as a reflective boundary.

The effective spatially averaged on-rate for particles on networks can be defined as . The effective long-time diffusivity on a dense network of semiflexible chains can be approximated as (see supplemental material for details):

where is the filament persistence length and the second term accounts for the limited persistence of particle directionality arising from bending of the tracks. Networks of random sticks lie in the limit , and the above approximation assumes that the length of each filament is long enough to not limit the run length (longer than ).

We compute the MFETs for pairs of particles on explicit networks, with varying on-rates . In figure 3, these encounter times are plotted against the effective diffusivity and are compared to matched simulations of run-and-tumble or purely diffusive particles. The run-and-tumble particles are assumed to initiate runs with the rate , run with velocity v, halt runs with the rate , and diffuse with diffusivity D until a new run is initiated. The purely diffusive particles are given a diffusivity of for comparison.

Figure 3. Encounter times for unbiased particles exhibiting different modes of motion. (a) Configuration of an explicit random-stick (top) and worm-like aster (bottom) network. (b) Mean first encounter time for pairs of: diffusive particles (yellow), run-and-tumble particles (orange), and particles on the specific network structures shown in (a) (pink and blue). Particles engaging in run-and-tumble motion or motion on explicit networks have short run lengths, with m. is varied by changing in the range . (c) Analogous plots of MFET for particles with long run lengths, m. Error bars show standard error of the mean. The radius of the domain is m, the particle diameter is m, the random stick network has filaments of length m and the worm-like aster has filaments of length m. Network density is m−1 in both cases. Particles have speed m s−1 in the running state and diffusivity in the diffusive state. Simulation details are provided in supplemental material.

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We first consider the regime in which the run length m is small relative to the critical value m. As shown in figure 3(b), such particles have similar MFETs regardless of their mode of motion or the network structure. The same similarity of short-length run-and-tumble motion to diffusive motion is also seen in figure 1. The presence of an explicit network reduces the encounter times slightly, an effect that can be attributed to particles being partially confined to the regions along individual filaments, requiring them to explore a total area smaller than the full domain before finding each other. This effect increases when particles spend more time on the filaments (higher , corresponding to higher ).

In the regime of longer run lengths (m), we observe that diffusive particles outperform both run & tumble particles and particles on explicit networks (figure 3(c)), particularly for high values of (high ) where the active particles spend most of their time running. This result is in keeping with the tendency of running particles to cover distances rapidly (high ) yet overshoot a small target, as seen in figure 1. Once the run length becomes a substantial fraction of the domain size, the higher value of does not actually increase the ability of the particles to search through the domain.

For particles that spend most of their time engaging in long runs (figure 3(c)), encounter times are slightly lower on a network composed of WLCs than on one composed of straight sticks. This effect can be attributed to the bending of the chains allowing running particles to sample a wider region of space as compared to perfectly straight runs. In other words, the volume of the ‘Wiener sausage’ (the area of space covered by a finite-sized particle undergoing a random walk) [68] is expected to be higher for a particle moving along a wiggling chain rather than a straight one.

Notably, encounter times of particles moving on predefined random-stick networks are similar to those engaged in spatially uniform run-and-tumble motion (figure 3(c)). For the physiologically relevant network density used here, particles encounter more than one filament during each diffusive phase of motion. The direction of each run is thus roughly uncorrelated to the previous one and the network can be well approximated by an isotropic continuum where particles can move in any direction after each tumble (or equivalently after each unbinding and rebinding cycle). Notably, the distribution of encounter times is approximately exponential for unbiased particles moving on an explicit network, as well as for the run-and-tumble particles (supplemental figure S2). Overall, these results suggest that unbiased organelle transport can be simplified to a run & tumble model with few parameters: an effective diffusivity, an effective run length, and an encounter radius.

We next consider encounter times between biased particles that move preferentially towards one end of each filament in an explicitly constructed spatial network. Each filament in the network has a ‘plus’ end and a ‘minus’ end. Structured, polarized networks are constructed by placing all filament minus ends at a central point (figure 4(a-i)) or at one of three separated points (figure 4(a-ii)) representing the MTOCs. The filaments are then grown following constrained WLC statistics to a length of m, with filaments truncated at the cell boundary in the 3-MTOC case (resulting in asymmetric accumulation of plus ends at the periphery, as observed in some animal cell types [35, 69]). Alternately, a random unpolarized WLC network is constructed by scattering minus ends uniformly throughout the domain and growing the filaments in an arbitrary initial direction to a fixed length of m (figure 4(a-iii)).

Figure 4. Encounter of biased particles on fixed network structures. (a) Example network structures for: (i) a WLC network with one MTOC and filament length 60 µm; (ii) a WLC network with three MTOCs and filaments extending to the domain boundary; (iii) a randomly scattered WLC network with filament length 60 µm. 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: , and .

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A particle is said to have bias b if, upon binding to a filament, it has a probability b of running towards the plus end. Thus, unbiased particles have b = 0.5, ones that always run towards the plus end have b = 1, and ones that always run towards the minus end have b = 0.

A key behavior observed for biased particles is their tendency to become ‘trapped’ in localized regions where multiple microtubules converge with an inward orientation. Such traps can be seen as peaks in the steady-state distribution of non-interacting particles (figures 4(b) and (e)). Modest trapping arises even in unstructured networks where filaments are scattered at random over the domain (the ‘random WLC’ networks in figure 4(a-iii)). The tendency towards trapping on random networks was previously shown to slow down the MFPT of biased particles to cross a rectangular region [49] or move from a centralized nucleus to the outer periphery of the cell, particularly when the traps were localized near the nucleus [34]. Here, we find that the presence of traps can actually speed up the encounter time between particles (figures 4(c) and (d)).

One limiting regime is a fully polarized network where all filaments originate with their minus ends attached to a single central point (MTOC). In this case, the MFET for particles biased to run towards the minus end (b = 0) is very short, since they quickly encounter each other at the MTOC (figure 4(c)). A bias towards the peripherally-oriented plus-ends of the network also slightly speeds up the encounter time as compared to unbiased particles. Particles moving towards the plus-end do not converge to a single point-like trap, but rather move rapidly towards a narrow region at the periphery of the cell. Within that boundary layer, they undergo random walks in the angular coordinate as they unbind, diffuse, and hop between the filaments. These results are analogous to those observed in spatially heterogeneous random velocity models, where the imposition of inward or outward bias in the bulk of the domain was also shown to speed up encounters [32]. The ability to find a partner is enhanced because the particles need only perform an effectively one-dimensional search in a narrow boundary layer, rather than a two-dime