Getting around the cell: physical transport in the intracellular world

Many biologically important proteins are embedded in cellular lipid membranes, including both the plasma membrane surrounding the cell itself and the much more extensive membranes of eukaryotic organelles [202]. Lateral diffusivities of membrane proteins have ranges of 6–10 μm² s⁻¹ in plasma membranes [203] and 0.2–0.5 μm² s⁻¹ in the ER membrane [204]. Confinement of a particle to a two-dimensional fluid membrane fundamentally alters its diffusivity in a manner dependent on the thickness and curvature of the membrane. A critical feature of a purely two-dimensional fluid is that hydrodynamic correlations do not decay but rather extend over the entire domain, leading to the famous Stokes paradox [205]. As a consequence, the size of the domain can be an important length-scale for determining the diffusivity even of very small particles far from the boundary. The classic Saffman–Delbrück model [206] derives the lateral diffusivity of a particle of radius a in a thin membrane of thickness h and viscosity μ_m, embedded within a bulk fluid with lower viscosity μ_s, as

$$\begin{equation}{D}_{\mathrm{SD}}=\frac{{k}_{\text{B}}T}{4\pi {\mu }_{\text{m}}h}\left(\mathrm{ln}\enspace \frac{{R}_{\text{corr}}}{a}-\gamma \right),\end{equation} \tag{ 4 }$$

where γ ≃ 0.6 is the Euler constant and R_corr gives an effective length-scale limiting planar hydrodynamic correlations. In contrast to free diffusion in a 3D solvent, this expression implies that lateral diffusivity on a membrane is only weakly dependent on particle size and is inversely proportional to the membrane thickness. The Saffman–Delbrück model is supported by in vitro experimental measurements [207], but requires significant alterations when the membrane is near a solid substrate [208], or when the protein radius is comparable to the membrane thickness [209]. The latter case, in particular, is relevant in the intracellular world, where typical membrane thicknesses (∼4 nm [210]) are comparable to protein dimensions.

The correlation length scale R_corr is defined by an interplay of several physical effects. In the case of a very large flat membrane domain, it is given by the Saffman–Delbruck screening length [206]: L_SD = hμ_m/μ_s, beyond which planar hydrodynamics are screened out by flows in the bulk fluid [208]. Alternately, it can be given by the overall extent of the membrane domain itself (R_mem; figure 7(a)), when this is smaller than the screening length [211]. The domain size R_mem is not well-defined for many biological systems. It may correspond to the size of membrane compartments with fixed boundaries defined by interaction with cytoskeletal filaments [212, 213]. In the specific case of particles diffusing laterally along a tubule-shaped membrane, it can be approximated as the radius of the tubule [214]. As a consequence, the lateral diffusivity of particles is expected to decrease with decreasing tubule radius, accounting for the experimentally observed slowing of diffusive spread on narrow reconstituted tubules [211].

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Mechanical properties of the membrane can also have an important impact on the lateral diffusivity of embedded proteins. For example, important physical effects arise when there is a mismatch between the preferred curvature of the embedded protein and the surrounding membrane curvature [215]. Alternately, many proteins show a mismatch between the length of the transmembrane region and the preferred thickness of the membrane [216] (figure 7(b)). In both cases, the mismatch engenders an elastic deformation field in the surrounding membrane [217, 218]. When multiple proteins come sufficiently close together for the deformation fields to overlap, they can experience attractive or repulsive forces mediated by the membrane elasticity [219]. Such interactions have a range of 1–2 nm for thickness deformations and 5–500 nm for curvature deformations [218].

As a result of these effects, membrane proteins diffuse across an effective potential energy landscape that can guide and modulate their motion. On a thermally fluctuating membrane, protein curvature preference has been postulated to enhance lateral diffusion by up to a factor of two, due to the attraction of the protein toward transient regions of matching curvature [220]. Both curvature and thickness preference can also attract proteins toward specific cellular regions. In particular, an energetic preference for membrane thickness has been implicated as a protein sorting mechanism in the secretory pathway [216, 221, 222], including capture at ERESs [55] and partitioning to secretion-bound lipid rafts in the Golgi [223, 224]. Similarly, curvature preference is believed to facilitate protein sorting into membrane tubules [225], the necks of budding vesicles [226], and the curved regions of dividing bacterial cells [227].

For particles diffusing within the bulk of the cell, a key assumption of the Stokes–Einstein relation (equation (3)) is that the cytoplasmic environment behaves as a purely viscous medium. This assumption has been challenged by a variety of studies that actively probe the rheological properties of the cytoplasm [232, 233], or else leverage 'passive microrheology'—visualizing and tracking the apparently passive trajectories of individual particles in live cells [36, 39, 228, 231, 234]. These studies are summarized in several excellent reviews on intracellular rheology [200, 235].

Passive particle-tracking microrheology enables explicit calculation of the MSD as a function of time (figures 8(a)–(c)), for comparison with the expected diffusive behavior described by equation (3). In some cases, injected beads or endogenous vesicles exhibit linear scaling of the MSD with time, as would be expected for a diffusing particle [26, 234, 236, 237]. More commonly, however, particle motion in cytoplasm is characterized as subdiffusive, with a sublinear scaling MSD ∼ t^α, where α < 1 [39–41].

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Subdiffusive scaling is expected when motion is driven by thermal fluctuations in a power-law fluid—a material with complex rheology, whose viscous and elastic moduli vary as a characteristic power law of the probing frequency [235]. For example, subdiffusive motion with α ≈ 0.75 is both theoretically expected and observed for particles embedded in gels of semielastic polymer filaments, such as F-actin [39, 238].

The usual physical model for passive particle movement in a viscoelastic fluid is termed 'fractional Brownian motion' [37, 200, 239]. This model derives from an overdamped generalized Langevin equation [240] featuring a power-law memory kernel K(t) which is convolved with the past time-course of particle velocities to give the drag force:

$$\begin{equation}\begin{aligned}\hfill \mu {\int }_{0}^{t}\enspace \mathrm{d}{t}^{\prime }K\left(t-{t}^{\prime }\right)\frac{\mathrm{d} \overrightarrow {r}\left({t}^{\prime }\right)}{\mathrm{d}t}& ={ \overrightarrow {F}}^{\left(\text{B}\right)}\left(t\right)\hfill \\ \hfill \left\langle {F}_{i}^{\left(\text{B}\right)}\left(t\right){F}_{j}^{\left(\text{B}\right)}\left({t}^{\prime }\right)\right\rangle & =\mu {\delta }_{ij}{k}_{\text{B}}TK\left(t-{t}^{\prime }\right)\hfill \end{aligned}\end{equation} \tag{ 5 }$$

where F^(B) is a Brownian force satisfying the fluctuation–dissipation relation and hence exhibiting the medium-dependent time correlations indicated above [241–244]. When the memory kernel is replaced by a delta-function, corresponding to an instantaneous relation between force and velocity as in a purely viscous fluid, the model reduces to classical Brownian motion. In a power-law fluid, the memory kernel is K(t) ∼ t^−α, effectively replacing the medium viscosity η with a frequency-dependent viscosity η(ω) ∼ ω^α−1 [200, 245]. Fractional Brownian motion gives rise to a sublinear MSD of passive particles [37]:

$$\begin{equation}{\left\langle {x}^{2}\right\rangle }_{\text{fBM}}=\frac{{k}_{\text{B}}T}{\mu }\frac{\mathrm{sin}\left(\alpha \pi \right)}{\pi \left(1-\alpha /2\right)\left(1-\alpha \right)\alpha }\enspace {t}^{\alpha }\end{equation} \tag{ 6 }$$

This model has been used to explain the observed subdiffusion of a variety of intracellular particles, including genomic loci [246, 247], mRNA molecules [248], and RNA–protein complexes [229, 247].

However, the MSD by itself cannot distinguish between several different models for subdiffusive motion [249]. For example, localization error [250] in tracking particle positions, crowding [251], confinement [252], or binding events with broadly distributed interaction timescales [253] can all give rise to MSDs with sublinear scaling. Other metrics have thus been developed to quantify the behavior of particles undergoing subdiffusive motion. One common metric is the velocity autocorrelation function (VACF), which tracks how velocities (defined by steps over different timescales δ) are correlated across a time-lag Δt. Namely, the velocity autocorrelation is given by

$$\begin{equation}{C}_{v}^{\delta }\left({\Delta}t\right)=\frac{1}{{\delta }^{2}}\left\langle \left[ \overrightarrow {r}\left({\Delta}t+\delta \right)- \overrightarrow {r}\left({\Delta}t\right)\right]\cdot \left[ \overrightarrow {r}\left(\delta \right)- \overrightarrow {r}\left(0\right)\right]\right\rangle .\end{equation} \tag{ 7 }$$

Unlike classical diffusion, where velocities are fully uncorrelated for all Δt > δ, fractional Brownian motion gives rise to negative velocity correlations that are self-similar across time-scales (figure 8(d)). Because several microrheology studies have shown similar behavior for the velocity autocorrelation of intracellular particles (figures 8(e) and (f)) [36, 229, 230, 246], the cytoplasm is often treated as a power-law fluid whose viscoelastic properties lead to fractional Brownian motion of passive components.

Brownian or fractional Brownian motion in a passive medium is driven by equilibrium thermal fluctuations. Thermally generated fluctuating forces must have a specific time-dependent correlation function determined by the rheology of the medium (equation (5)). The interior of a living cell, however, is an environment that is manifestly outside the equilibrium regime, with fluctuations driven by a wide array of active energy-consuming processes with different underlying temporal correlations.

A plethora of recent experimental evidence has shown that even apparently diffusive particle dynamics rely on active cellular processes and are not driven primarily by thermal fluctuations [254]. Active microrheology measurements can be used to probe the force-response dynamics of the cytoplasm by directly controlling the forces applied to beads caught in optical and magnetic traps. Such measurements tend to indicate that the cytoplasm responds to force as a largely elastic material, in direct contrast with the apparently diffusive motion of passive particles [232, 255]. Attenuation of active cellular processes (e.g.: by ATP depletion) results in severe reduction in the mobility of cytoplasmic particles [26, 232, 256]. Furthermore, the temperature dependence of apparent particle diffusivity inside the cell is non-linear, in contrast to expected behavior for generalized diffusive motion (equation (5)). Instead, the temperature dependence is Arrhenius-like, with mobility scaling according to D ∼ exp(−E_a/k_BT) as expected for reaction rates of activated processes [257].

A number of different active processes are believed to play a role in the apparent particle diffusivity inside the cell. Myosin motor activity been shown to contribute substantially to overall particle mobility in mammalian cytoplasm [232, 256, 258, 259]. Inhibition of directed motor-driven transport is also known to reduce active diffusivity of apparently passive organelles [26]. Recent evidence indicates the diffusivity of individual active enzyme molecules can be significantly enhanced in the presence of their substrates, through mechanisms that are currently unclear [260–262].

The behavior of particles driven by active fluctuations is determined by the spatiotemporal correlations of forces acting on the particles $\left[{ \overrightarrow {F}}^{\left(a\right)}\right]$ and the memory kernel (K) describing medium response. If the active forces have correlation $\left\langle {F}^{\left(a\right)}\left(t\right){F}^{\left(a\right)}\left({t}^{\prime }\right)\right\rangle \sim \vert t-{t}^{\prime }{\vert }^{-\beta }$ and the memory kernel scales as K(t − t') ∼ |t − t'|^−α, then the MSD of the particle is given by [263]

$$\begin{equation}\text{MSD}\sim {t}^{2\alpha -\beta }.\end{equation} \tag{ 8 }$$

For a particle pushed by a purely processive force β = 0, while forces with delta-function correlations correspond to the limit β → 1. Linear scaling of the MSD arises for particles undergoing thermal diffusion in a purely viscous medium (α = β = 1). Alternately, it can also arise for particles in a purely elastic medium (α = 0) pushed by random processive forces that themselves accumulate as a random walk over time (β = −1). The latter model has been proposed for movement driven by an accumulation of actomyosin contraction events, over timescales shorter than the processivity time of an individual myosin motor [232]. In the interest of brevity, regardless of the underlying physical cause, we will refer to stochastic particle movements with negligible processivity as apparently diffusive in the remainder of this manuscript.

An assumption of the Stokes–Einstein relationship for diffusing particles (equation (3)) is that the particles are embedded in a continuous medium. The interior of a eukaryotic cell is inherently very crowded, with proteins constituting over 20% by mass of mammalian cell cytoplasm [264]. In addition, organelle structures ranging from vesicles to reticulated tubules and cytoskeletal networks are interspersed throughout the cell, occupying 40%–50% of cell volume [210]. In most models of particle motion within the cytoplasm, these crowding agents are averaged out to yield an effective viscous or viscoelastic medium. However, this approximation can lead to inaccurate predictions for transport behavior in the cytoplasm. For instance, the dependence of the diffusion coefficient on particle size (D ∼ R in a continuum fluid) is highly non-linear, with nanoscale proteins typically experiencing an effective viscosity that is orders of magnitude lower than that measured with micron-sized beads or vesicles [265]. Furthermore, protein complexes sized in the tens of nanometers tend to exhibit purely diffusive motion [266], rather than the subdiffusive behavior observed with vesicle-sized probes that are an order of magnitude larger [40, 41, 228]. This strong dependence of medium properties on probe size is generally found in gels, where particles much smaller than the pore size move freely through the gel while larger particles rely on rare jump events or large-scale rearrangements to move between pores [267, 268]. For proteins embedded in the plasma membrane, the actin cortex has also been shown to form a meshwork of obstacles that reduces effective diffusivity, particularly for larger probes [269].

In addition to individual crowders of all shapes and sizes, broadly-distributed spatial heterogeneity has been shown to play an important role in governing Brownian motion of cytoplasmic components. Quantification of individual step size distributions for RNA–protein particles [229], colloidal tracers [249, 270], and membrane-bound receptors [271] indicates that they do not follow a Gaussian distribution as would be expected for thermally diffusing particles in a uniform viscous or a viscoelastic medium (figures 8(g)–(i)). Instead, the step sizes have a Laplace distribution, with probability density P(Δx) ∼ exp[−Δx/λ(t)]. This scaling is indicative of a breakdown of spatiotemporal homogeneity in the particle motion, which would imply by the central limit theorem that each time-step should involve the sum of many uncorrelated displacements and should thus follow a Gaussian distribution. Similar long-tailed distributions of step-sizes are observed for the dynamics of tracers in a suspension of active swimmers [272], in glassy systems [273], and in polymer solutions (figure 8(h)) [231, 274].

The origin of such distributions has been attributed to broadly distributed diffusivities of individual particles caught in different regions of a heterogeneous environment [229, 275, 276], with exponential distributions of the diffusion constant giving rise to the observed Laplace distribution in stepping times. Indeed, diffusion coefficients extracted from individual trajectories of intracellular particles generally exhibit very broad distributions that are not strongly peaked around a preferred value [27, 247, 277]. In some cases, this observation has been attributed directly to local variations in the density of obstacles formed by organelle structures such as the ER [278].

More sophisticated models of 'diffusing-diffusivity' incorporate time correlations as the particle moves through the heterogeneous environment, with D(t) itself treated as a time-dependent random variable [276, 279, 280]. Beyond a characteristic correlation time, such a particle samples over many diffusivities and its step-size distribution again begins to look Gaussian (figure 8(g)), as has been observed in some experimental measurements [231, 280].

Overall, the broad non-Gaussian distributions of step sizes over commonly measured time-scales highlight the heterogeneity of the intracellular medium and the difficulty of making general conclusions based on 'typical' particle diffusivities.

In addition to macromolecular crowding, the diffusion of many intracellular particles is limited by confinement in subcellular regions of complex geometry. Subcellular morphology is diverse, including shapes resembling spheres, tubes, sheets, labyrinths, beads on a string, and networks. Tubes and sheets are particularly common, and effectively confine diffusion to one or two dimensions, respectively. Here we outline the effect of these morphologies on the diffusive spreading of proteins confined within organelles.

The ER and mitochondria are ubiquitous intracellular structures that exemplify several of these morphologies. Both of these organelles can form extensive tubular networks (figures 9(a) and (b), left) [288–290] or dynamically break up into globular structures [291, 292] (restricted to certain stress or perturbative conditions in the case of the ER [293, 294]). The ER also forms stacks of flat membranous sheets in the perinuclear region (figure 9(a), bottom right). The morphology of these organelles is altered in different cell types [54, 295], growth conditions [296, 297], cell cycle stages [298, 299], or states of stress [300, 301]. The complex geometries of both ER and mitochondria are believed to be linked to their functional roles in the cell [290, 302].

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Confinement within stable tubular geometries is found in mitochondrial [303], ER [290], and peroxisome [304, 305] networks, as well as bead-on-a-string structures formed by nuclei in certain cell types such as human leukocytes [306]. Transient tubules are also observed during vesicle budding and organelle fission [307], ER-to-Golgi transport [63], and peroxisome division [308, 309]. Tubule radii can range from ∼10 nm for dynamin-constricted regions [310] to ∼300 nm for mitochondrial network tubules [303]. For membrane proteins, models of diffusion on curved surfaces have shown that confinement to increasingly narrow tubules leads to slower spreading over the surface even when the diffusion constant and membrane surface area are kept constant [311]. This purely geometric effect is thought to arise from the local curvature and global topology of tubular membranes. Crowding of proteins on tubular membranes can lead to additional effects, including effectively anisotropic diffusion in the lateral versus circumferential directions [312].

For proteins in the lumen of a tubule, variation in tube radius can give rise to an entropic effect wherein locally narrower regions serve as effective diffusion barriers [313]. The local axial diffusivity in a tubule of heterogeneous radius R(x) is given by the Fick–Jacobs equation:

$$\begin{equation}D\left(x\right)=\frac{{D}_{0}}{{\left[1+{R}^{\prime }{\left(x\right)}^{2}\right]}^{\alpha }},\end{equation} \tag{ 9 }$$

where D₀ is the diffusivity in free space, α = 1/3 for two dimensions and α = 1/2 for three dimensions [313, 314]. An extreme case of entropic traps can be seen in geometries with narrow-necked regions branching away from a main tubule, as in dendritic spines (figure 9(d)). These traps serve as effective obstacles to diffusive motion along the tubule, resulting in a reduced diffusivity at long times (when many traps have been sampled) and anomalous diffusion at intermediate times [243, 315].

Confinement within complex organelle geometries gives rise to a discrepancy between the actual domain explored by a particle and its apparent motion in the 3D space where the organelle is embedded. For instance, particles on a curved membrane surface generally traverse longer lengths than the Euclidean distance between a start and end point (figure 9(e)), leading to underestimation of diffusivity when 3D spreading is analyzed [287]. Particles confined to a reticulated network of tubules are restricted to move along one dimension between each neighboring node. This effect decreases long-range diffusivity by a factor of 2 or 3 for a fully connected regular planar or 3D lattice, respectively (figure 9(f)). Comparison of fluorescence recovery after photobleaching experiments with simulations on extracted ER structures suggests that diffusive recovery times in the ER are 1.8–4.2× longer than would be expected from local diffusivity measurements [316]. Furthermore, the convoluted geometry of this organelle seems to have a greater effect on membrane than on luminal proteins. Simulations on realistic tubular ER geometries indicate that membrane proteins explore ER regions up to 4× slower than luminal ones, even when the diffusion coefficient is identical for both [317].

The effect of complex confining geometry or occluding barriers on long-range particle diffusion has been extensively explored in the context of transport through porous media and over spatial networks [318]. These effects are often described via an emergent quantity called 'tortuosity'—which is conceptually defined as the ratio between the typical length traversed by a diffusing particle and the Euclidean distance between its start and end points (figure 9(e)) [319]. A common simplifying model for environments with high tortuosity is that of percolation on a lattice [320]. The medium is represented as a lattice network with randomly removed edges. Such systems exhibit a phase transition when the fraction of remaining edges reaches a critical value p_c, below which the network becomes disconnected and particles can no longer penetrate throughout the domain. Percolation systems exhibit a number of universal scaling behaviors, including a slow-down in effective diffusivity according to

$$\begin{equation}D\sim {\left(p-{p}_{\text{c}}\right)}^{\mu }\end{equation} \tag{ 10 }$$

as the fraction of remaining edges approaches the critical value [321]. The scaling exponent is μ = 1.3 for two-dimensional and μ = 2.0 for 3D lattices [322].

Within cellular organelles, the connectivity of the space available for diffusion is determined by the overall organelle geometry, as well as the presence of intra-organelle substructures that serve as obstacles for mobility. The perinuclear ER presents an example of sheet geometry in the form of stacks of flat parallel cisternae bounded by membrane sheets [290]. The cisternae are interconnected with ramp-like spiral dislocations reminiscent of a parking garage [283] (figure 9(a), bottom right). The morphology of this structure is thought to directly modulate diffusion, with the ramps substantially enhancing diffusive transport between stacked cisternae when compared to individual holes in the membrane [323]. In particular, the unique connection geometry allows diffusing particles to transition between sheets by spiraling around dislocations rather than searching for small holes serving as localized connections between flat sheets. Even in the presence of these spiral structures, the limited connectivity of stacked ER sheets results in an effective perpendicular diffusion that is roughly 10-fold slower than local diffusivity [323].

An example of labyrinthine structures that limit connectivity within organelle compartments are the mitochondrial cristae—convoluted folds of inner mitochondrial membrane that occlude much of the mitochondrial matrix space (figure 9(b), right) [324]. There are about 6–8 cristae per μm of mitochondrial length, each of which serves as an impenetrable barrier to the diffusion of molecular species [325]. Early studies indicated that these protrusions must stretch across nearly the entire mitochondrial cross-section in order to have a substantial impact on diffusivity [326]. Simulations of diffusive spreading in the presence of multiple such overlapping barriers show that cristae are expected to slow long-range axial diffusion of matrix proteins by a factor of 5–6 [325]. Effective axial diffusion of proteins embedded in the inner mitochondrial membrane may also be slowed by up to an order of magnitude by the convoluted morphology of these structures [327].

In the outer segment of mammalian photoreceptor cells, flat lammellar disc membranes form similar occlusions, leading to a high tortuosity for axial transport (figure 9(c)) [328, 329]. Axial diffusivity in this compartment has been measured as roughly 50-fold slower than the nearby inner segment compartment, with a factor of 20–40× accounted for by the increased tortuosity due to membrane occlusions [329].

In the preceding discussion we addressed the impact of various physical factors on diffusive particle motion. Here, we consider the interplay of diffusion and morphology in limiting the kinetics of intracellular encounters and reactions. For freely diffusing particles in a 3D continuum, maximal reaction rates are proportional to particle concentrations. The steady-state current of particles to a perfectly absorbing spherical target of radius a is given by

$$\begin{equation}J=4\pi Da{c}_{0},\end{equation} \tag{ 11 }$$

where D is the particle diffusivity, c₀ the bulk particle concentration [333], and J represents the rate of particles arriving at the target. However, this linear relationship between concentration and reaction rate does not necessarily hold when particles are confined to complex geometries or embedded in domains of reduced dimensionality [318, 334].

Target search processes involving randomly moving particles fall into two broad categories: compact and non-compact [318, 335, 336] (see figure 10(a)). In a compact search process, a particle will cover most of the sites within each subregion it visits. Such particles generally find the target after comprehensively exploring a finite subsection of their domain, and their target search times strongly depend on their starting position [337]. By contrast, a non-compact search sparsely samples subregions of the domain, will generally reach the domain boundary before finding the target, and has search times largely independent of starting position.

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For random walks on self-similar (i.e. fractal) geometries, the behavior of the search process is determined by two key dimensions. The dimensionality of the random walk itself (d_w) can be defined by the scaling of MSD with respect to time (in the absence of confinement): $\text{MSD}\sim {t}^{2/{d}_{\text{w}}}$ [318]. Equivalently, d_w describes the scaling between the time to exit a sphere and the sphere size R: ${t}_{\text{exit}}\sim {R}^{{d}_{\text{w}}}$ [338]. The fractal dimension (d_f) describes the dimensionality of the medium within which the walk is embedded, relating the number of sites (N) with the spatial extent of a region (R) according to $N\sim {R}^{{d}_{\text{f}}}$ [318]. Compact search corresponds to the regime where d_w > d_f, such as canonical diffusion (d_w = 2) on a one-dimensional line (d_f = 1). The opposite regime (d_w < d_f) is termed non-compact search and includes canonical diffusion in three dimensions (d_f = 3).

The mean time ⟨T⟩ for a randomly moving particle to find a target site in the fractal medium is then given by the following scaling laws with respect to the domain volume N and initial distance from the target r [338]:

$$\begin{equation}\langle T\rangle \sim \begin{cases}N\left(A-B{r}^{{d}_{\text{w}}-{d}_{\text{f}}}\right),\;\text{for}\;{d}_{\text{w}}{< }{d}_{\text{f}}\;\left(\text{non}-\text{compact}\right)\quad \hfill \\ N\left(A+B\enspace \mathrm{ln}\enspace r\right),\;\text{for}\;{d}_{\text{w}}={d}_{\text{f}}\quad \hfill \\ N\left(A+B{r}^{{d}_{\text{w}}-{d}_{\text{f}}}\right),\;\text{for}\;{d}_{\text{w}}{ >}{d}_{\text{f}}\;\left(\text{compact}\right)\quad \hfill \end{cases} .\end{equation} \tag{ 12 }$$

In the case of non-compact search, the dependence on starting position disappears for sufficiently large r, and the search time is simply proportional to the system volume, as expected for classical 3D kinetics (equation (11)). The distribution of search times in this case exhibits an exponential drop-off with a single characteristic time-scale corresponding to the average search time [335]. By contrast, the compact case results in 'geometry-controlled' kinetics, with a search time that depends strongly on starting position, even for initially distant particles. In this situation, the distribution of search times exhibits decay over a range of different time-scales whose breadth depends on the dimensions d_f and d_w [335]. The mean search time, averaged over all starting positions, scales as $\overline{\left\langle T\right\rangle }\sim {N}^{{d}_{\text{w}}/{d}_{\text{f}}}$, indicating that the slowing of kinetics with increased volume is super-linear [335]. It should be noted, however, that the broadly distributed reaction times in compact systems are not well-described by this single mean first-passage time [339]. For particles undergoing unhindered canonical diffusion (d_w = 2), the overall reaction rate for a particle to find any stationary target (defined by $k{:=}1/\overline{\left\langle T\right\rangle }$) is expected to scale as follows depending on the dimensionality of the confining domain [337, 340]:

$$\begin{equation}k\sim \begin{cases}^{2},\quad \;\left(\text{1D}\right)\quad \hfill \\ c\enspace \mathrm{log}\enspace c,\quad \;\left(\text{2D}\right)\;\quad \hfill \\ c,\quad \;\left(\text{3D}\right)\quad \hfill \end{cases},\end{equation} \tag{ 13 }$$

where c is the target concentration (or the inverse of the volume per target).

The impact of confinement geometry on target search is particularly relevant for molecules that must find sparsely scattered binding partners within an organelle. This includes, for example, newly-translated secretory proteins searching for an exit site within the ER network [341], or mitochondrial matrix proteins searching for nucleoids [342]. While realistic cellular structures are not true fractals, similar considerations of compact versus non-compact search processes can be applied to understand the effect of organelle morphology on kinetics. For example, calculation of diffusive first-passage times to find one of many point-like targets on planar ER networks extracted from mammalian cell images indicate that the search domain transitions from effectively 2D to effective 1D with increasing concentrations of the target sites (figure 10(b)). Due to the compact nature of this process, the rate at which proteins find punctate exit sites in the ER is expected to scale super-linearly with exit site density.

One important class of target-search processes, known as 'narrow escape' problems, consists of particles that must find their way to a very small region on the boundary of their confining domain. This class of problems encompasses molecules that need to exit specific cellular regions, such as ER proteins moving from cisternae to peripheral tubules [290] or reaching an exit site for export [55], signaling factors leaving dendritic spines [286], or mRNA encountering nuclear pores [343] (see figure 10(c)). It also includes reactions with a fixed target on the membrane of an organelle within which the searcher is confined. The mean first-passage time for a diffuser to reach a narrow target whose area covers a small fraction () of the boundary can be approximated as:

$$\begin{equation}\begin{aligned}\hfill {\tau }_{\text{MFP}}& \simeq \frac{{R}^{2}}{D}\left[\mathrm{ln}\enspace \frac{1}{{\epsilon}}+\mathcal{O}\left(1\right)\right]\enspace ,\quad \left(2D\right)\hfill \\ \hfill {\tau }_{\text{MFP}}& \simeq \frac{V}{4{\epsilon}D}\left[1+\frac{{\epsilon}}{\pi }\enspace \mathrm{ln}\enspace \frac{1}{{\epsilon}}+\mathcal{O}\left({\epsilon}\right)\right]\enspace ,\quad \left(3D\right)\hfill \end{aligned}\end{equation} \tag{ 14 }$$

for a circular or a spherical domain, respectively [344–346]. The case of a particle trapped in a short cylinder lies intermediate between the two regimes, transitioning from two- to three-dimensional as the height of the cylinder increases [347]. This geometry can be particularly relevant for target search by particles trapped between flat sheets, as in the ER cisternae or lamellar discs of photoreceptor cells.

A common model for diffusion in the presence of obstacles or in reticulated or porous structures is to treat the process as a series of hops between compartments that are themselves rapidly equilibrated [330, 348]. Such geometries can result in a substantial reduction in long-range diffusivity without a concomitant decrease in the reaction rate [330]. Interestingly, the connectivity of compartments can be tuned in such a way that diffusive particles propagate in a wave-like manner, with transient concentration peaks appearing in different containers [348].

The nature of a target-search process in compartment networks is determined by the dimensionless parameter x = DL/D₀a, where D₀ is the diffusivity within a compartment, D the long-range effective diffusivity, L the compartment size, and a the particle reaction radius. The reaction rate exhibits one of two possible behaviors [330]:

$$\begin{equation}\begin{aligned}\hfill k& =4\pi \left(1-{P}_{r}\right)DL,\quad x\ll 1\hfill \\ \hfill k& =4\pi {D}_{0}a\left(1-\frac{a}{L}+\frac{aD}{L{D}_{0}}\right),\quad x\gg 1,\hfill \end{aligned}\end{equation} \tag{ 15 }$$

where P_r is the probability of returning to an already-sampled compartment. When x < 1, the process is compact and each compartment is fully explored as the particle moves through the medium (figure 10(a)). By contrast, for x > 1, the search process is sparse and the particle typically encounters the target only after multiple visits to the compartment containing the target. In this regime, when the target size is much smaller than the compartment, the long-range diffusivity may be greatly reduced (D ≪ D₀) without significantly changing kinetic rates. For enzyme diffusion in the cytoplasm, estimated pore sizes are roughly 10 times bigger than the protein size [264, 349], implying that the sparse search regime is relevant for cytoplasmic kinetics.

The effect of macromolecular crowding on reaction rates can be approximated in an analogous manner by treating reactants as moving between crowder-free cavities [350]. It should be noted that non-specific binding to reactants, and finite local reaction rate upon encounter can further slow the overall reactive flux in the presence of crowding [350]. Once interacting molecules are coincident in space, they must also find the correct relative rotational orientation for binding or activity [351]. Molecules coming together will typically experience many 'microcollisions', allowing time for reorientation through random chance or intermolecular interactions that favor alignment [351]. Effective confinement from crowding cavities provides further opportunity for sites to align and a reaction or binding event to occur.

For particles diffusing on a network, the connectivity of compartments (or nodes) plays an important role in regulating target search times, as well as large-scale diffusivity [352]. For reticulated structures similar to those of the peripheral ER or mitochondrial networks, target search times were recently shown to be determined largely by the total network edge length and the loop (or cyclomatic) number [281, 289]. Loop number is a global metric of connectivity, defined by Γ = N_e − N_n + 1 where N_e is the number of edges and N_n is the number of nodes. The parameter corresponds to the number of independent cycles in the network structure [353]. Increasing loop number decreases search times, while increasing edge length increases them, with a scaling relationship that can be derived from the slowed diffusivity on a percolation lattice (equation (10)) [281]. A recent study on yeast mitochondrial networks demonstrated that network connectivity can be altered by mutations in specific proteins responsible for mitochondrial fusion and fission [289]. Simulations of diffusive search over these network structures indicate that the reduced connectivity in mutant networks is expected to slow encounter times by almost two-fold for particles at low concentrations [289].

Diffusive transport inside cells is modulated by the mechanics of intracellular media, by active non-thermal fluctuations, and by the presence of obstacles and complex subcellular geometries. These physical factors control both the overall dispersion and the rates of encounter between particles. Diffusive transport thus provides a physical link between the morphology and dynamics of cellular structures and the kinetics of biomolecular reactions that underlie cell function.

The basic components of motor-driven transport include the cargo itself, the motor proteins, a variety of adaptor proteins and linkers that attach motors to the cargo, and the cytoskeletal filaments that serve as a substrate for walking motors (figure 11). Both actin filaments and microtubules can serve as highways for motor-driven transport. Both are polarized, with distinct '+' and '−' ends, governing the direction of motor movement. In plant cells, the motion of a variety of myosin motors along polarized actin filaments is responsible for long-range cargo delivery, as well as the establishment of persistent cytoplasmic flows [365]. In animal cells, the myosin-V motor has been shown to contribute to local organelle positioning in actin-dense cortical regions [366–368]. However, long-distance transport in animal cells primarily occurs along microtubule highways.

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Microtubules (MTs) form long hollow tubes, consisting of 13 parallel protofilaments, with motor proteins attaching to the outside of the tube. Interestingly, diffusive transport in the hollow interior of a microtubule has also been shown to play an important role in the spread of several microtubule-modifying proteins [369, 370]. Microtubules are quite stiff, with effective in vivo persistence lengths on the order of 30 μm, enabling them to fluctuate around relatively straight configurations on typical cellular scales [371]. In many animal cell types, they are organized with their minus ends anchored near the nucleus and their plus ends extending toward the cell periphery. Microtubules are highly dynamic, undergoing cycles of growth and depolymerization that allow for rapid remodeling of the transport highway network [372], as well as bending and sliding events that contribute to cargo motion [373].

Two families of motor proteins execute transport along microtubules. The kinesin superfamily [374] is generally responsible for anterograde transport: movement toward microtubule plus ends, which often corresponds to the direction away from the nucleus. Dynein motors drive retrograde motion toward microtubule minus ends [375, 376]. Both types of motors form protein complexes with two ATP-burning motor domains that bind to the microtubule, linked to a long tail that attaches to cargo, often via an adaptor complex [18]. The motors walk in a hand-over-hand fashion, with some (e.g.: kinesin-1) following individual protofilaments while others (including kinesin-2 and dynein) undergo frequent side-stepping to neighboring protofilaments [377, 378]. The mechanochemical behavior of individual molecular motors has been extensively explored at the single-molecule level in vitro [379, 380]. In a living cell, many motors can attach to each cargo, and their cooperative behavior determines the speed, processivity, and direction of cargo motion [19, 381–383].

Specialized adaptor proteins control the complement of motor molecules recruited to a particular cargo [20, 375]. These adaptors make it possible for a wide range of cargos to be transported by a limited variety of motor proteins, as well as controlling the direction and processivity of motion [376, 384–386]. In general, adaptor proteins are bound directly by receptors on the cargo surface, by both kinesin and dynein motor complexes, and by a variety of signaling proteins that serve to activate or repress transport [20].

As an alternative to the direct recruitment of motors via an adaptor protein, some cargos have been found to engage transiently with other motile organelles, moving by a non-canonical form of motor-driven transport termed 'hitchhiking' [9, 26, 167, 387–392]. In place of an adaptor protein, a linker protein attaches the hitchhiking cargo to a carrier organelle, which connects through an adaptor protein to the motor. Specific linker proteins have been identified for several hitchhiking cargos [9, 168, 392], and the density, length, and stiffness of these linker proteins can serve to modulate the efficiency of the hitchhiking interaction [393]. Both linker proteins and adaptors share the common feature of enabling specific control of transport for a particular cargo, without affecting the movement of other cellular components.

The direction and processivity of transport along a single microtubule can vary widely for different cellular systems. Some cargos, such as post-Golgi synaptic precursor vesicles in proximal regions of neuronal axons, move primarily in the anterograde direction toward the cell periphery [397–399]. Others, such as endocytic vesicles carrying growth factor signals [400] and neuronal autophagosomes [166], move primarily in the retrograde direction, toward the cell nucleus. Many cargos are bidirectional, exhibiting both types of motion with varying run-lengths prior to switching directions [19, 165, 401, 402]. At one extreme of highly processive bidirectional motion lie mitochondria in neuronal axons, which move for many tens of microns in either anterograde or retrograde directions, undergoing pauses of varying duration, but rarely reversing their direction after pausing (figure 12(a)) [394]. By way of contrast, lipid droplets in Drosophila embryos [403], lysosomes in neurons (figure 12(b)) [382], as well as endosomes and hitchhiking peroxisomes in fungal hyphae (figures 12(c) and (d)) [26, 392] all switch directions frequently, with typical run-lengths of about 0.3–10 μm. Cytosolic proteins engaged in slow axonal transport have been observed to exhibit even shorter processive runs of about 0.1 μm, thought to arise from transient interactions with passing cargos [106].

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The direction and run-length for a motor-driven cargo moving along a single microtubule is thought to be determined by the complement of associated motors, as well as regulatory modifications to motors, adaptor proteins, and the microtubules themselves (figure 13). Cargos that exhibit bidirectional motion are generally attached to both kinesin and dynein motors simultaneously [382, 404–406]. Even axonal mitochondria and autophagosomes, with their very long processive run-lengths, have been shown to carry both kinesin and dynein motors regardless of whether they are stationary or moving in the anterograde or retrograde direction [166, 407, 408]. The question of how multiple motors coordinate to determine the direction, speed, and processivity of cargo has been the topic of much theoretical and experimental work over the past two decades.

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The classic model for opposing motor interactions is a 'tug-of-war' between multiple motors that come on and off the microtubule stochastically and pull in their characteristic direction when engaged (figure 13(a)), with the overall direction of movement dictated by the net generated force [381, 409]. When coupled with experimental measurements of the number of motors on a cargo and the force-response parameters of individual motors, the tug-of-war model can quantitatively recapitulate aspects of in vivo bidirectional motion for vesicles in mammalian neurons [382] and endosomes in Dictyostelium slime molds [410], as well as multi-motor assemblies in vitro [411].

However, this simple model fails to account for a number of puzzling observations indicating cooperative rather than competitive behavior between kinesin and dynein motors on the same cargo [19, 383]. Qualitatively, the presence of both kinesin and dynein motors has been found to be necessary to activate motion in both anterograde and retrograde directions [412–414], raising the so-called 'paradox of co-dependence' [19]. Quantitatively, a thorough parameter scan for the tug-of-war model has shown that no variant of the model can simultaneously reproduce the in vivo distribution of run-lengths and pausing behavior of bidirectionally motile lipid droplets [415, 416].

A number of mechanisms for positive cooperativity between opposing motors have been proposed as an alternative to the antagonistic tug-of-war model [19]. One possibility is the existence of direct biochemical and mechanical interactions wherein one motor type serves to activate the other or to push it out of an auto-inhibited state [406, 414, 417]. An alternate mechanism relies on inactive motors entering a weakly-bound diffusive state wherein they function as tethers that prevent cargo dissociation from the microtubule and hence increase processive run-lengths driven by the dominant active motor [19]. Such an effect may account for the increased processivity of kinesin-carried cargos along microtubules in the presence of myosin-V motors, and the reciprocal increase in myosin-V processivity on actin filaments in the presence of kinesin [418, 419].

Cooperation between multiple motors pulling in the same direction has also been proposed to enhance the speed and processivity of transport. In vitro measurements on reconstituted systems show that the presence of multiple kinesin motors allows for longer run lengths and larger stall forces [420, 421], with similar cooperative effects observed for multiple dyneins [422]. Furthermore, coupling of many kinesins bound to a fluid lipid membrane has been shown to increase cargo transport velocity without altering the behavior of individual motors [423]. Theoretical studies of load-sharing between motors help clarify the importance of key mechanical parameters in determining motor cooperativity, as well as highlighting the limitations of purely mechanical models and the need to incorporate biochemical coupling effects [383, 424–426].

In addition to interactions between the complement of motors attached to a cargo, processive motion along a microtubule can also be regulated by external signals targeting motors and adaptor proteins [427] (figure 13(b)). These signals often take the form of a biochemical modification through a signaling pathway that responds to the local intracellular environment or the state of the cargo itself. For example, calcium ion binding to the mitochondrial adaptor complex consisting of Miro and Milton proteins results in transient halting by dissociation of kinesin motors from the microtubule [165, 407]. Similarly, a byproduct of glucose metabolism serves as a substrate for modifying the Milton adaptor protein, inhibiting mitochondrial motility [171]. By coupling transport behavior to the local biochemical environment, these pathways can result in targeted localization of mitochondria to regions with high metabolic demand [48, 184, 428] or high glucose supply [171, 185] within extended neuronal projections.

A permanent cessation of mitochondrial transport can also be triggered through the PINK1/Parkin pathway, which is activated when the mitochondrial membrane potential (a marker for mitochondrial health) drops too low and results in the degradation of the Miro adaptor protein [165]. A cell can thus precisely control the positioning of its mitochondria in response to local cytoplasmic conditions and mitochondrial health. Another example of organelle state modulating transport behavior can be seen in neuronal autophagosomes, whose biochemical maturation is coupled to their transition from bidirectional motion near sites of synthesis at distal axonal tips to robust retrograde motility toward the cell body [166].

Microtubules themselves can serve as a substrate for post-translational modifications and other signals that regulate transport processivity [427, 429, 430] (figure 13(c)). MAPs bind to the external surface of microtubules and differentially regulate motor protein behavior. For example, tau proteins tend to cause kinesin detachment at low concentrations with little effect on dynein [431]. Gradients of tau proteins (which have been observed in neuronal axons [432]), can thus be used to tune the anterograde or retrograde bias, as well as processivity, of cargo transport [421, 433–435]. Other MAPs differentiate the microtubule-binding affinity of separate types of kinesin motors, allowing kinesin-3-bearing cargos to be sorted into dendritic projections while those carrying kinesin-1 are relegated to the axons [436].

Even in the case where a cargo follows a single microtubule or polarized bundle, its direction and run-length are thus a complicated function of the complement of attached motors, the decoration of the microtubule track, and the spatial profile of signaling molecules that inhibit transport.

The processive motion of a motor-driven cargo along a microtubule is inherently limited by the crowded environment within a living cell. Crowding by filamentous macromolecules gives rise to a viscoelastic rheology of the cytoplasm (section 4.1.2) which results in size-dependent and time-dependent drag forces experienced by the moving cargo. As a consequence, in vivo movements of cargo tend to be 'bursty', with speed fluctuations consistent with a slow build-up and rapid release of mechanical stresses [437, 438]. Models of motor-driven motion which incorporate complex fluid rheology predict the emergence of an anomalous transport regime with superdiffusive yet sub-ballistic scaling of the mean-squared displacement (MSD ∼ t^α with 1 < α < 2) [439, 440]. In reconstituted in vitro systems with a viscoelastic medium, increased densities of filamentous crowders have been shown to drastically reduce the transport velocity of cargos carried by teams of kinesin motors [441].

In addition to altering the rheology of the cytoplasmic medium, crowded conditions within the cell imply the ubiquitous presence of obstacles, both directly bound to the microtubule track and in the cytoplasm at large [442]. Individual molecular motors vary in their ability to bypass MAPs that serve as roadblocks along the transport highway (figure 13(c)). Single kinesin-1 motors generally dissociate when encountering a road-block, though teams of such motors can effectively bypass the obstacle [377]. Individual dynein motors, on the other hand, are much more capable of side-stepping to neighboring protofilaments, allowing them to successfully bypass microtubule-bound obstacles [377, 443]. The increased ability to maneuver around obstacles afforded by the presence of different motor types has been proposed as a key evolutionary advantage to bidirectional motion [444].

When encountering large obstacles, such as other vesicles attached to the same track or intersecting microtubules (figure 13(d)), 3D motion of the cargo around its track is required for maneuvering around the obstacle [445]. In vivo tracking of anisotropic particles indicates that 3D rotation of the cargo occurs during long pauses that result in directional reversals on the same microtubule or a nearby parallel track [446]. These pauses were postulated to arise from obstacle encounters, with release and engagement of alternate motors allowing the cargo to bypass the obstacle. When encountering a microtubule intersection, cargo can also switch to the intersecting microtubule, reverse, or pass by it, in a manner dependent on the geometry of the intersection [447] and the complement of attached motors [356, 448]. The extent to which bypassing of an intersection in vivo involves side-stepping of individual motors versus switching or tug-of-war behavior between multiple motors remains largely unknown [445].

An additional source of transport obstacles comes from traffic jams formed by individual molecular motors bound to and moving along microtubules (figure 13(e)). These traffic jams can be described by the classic physical model of a 'totally asymmetric simple exclusion process' [449], which consists of non-intersecting particles moving along a line and predicts the onset of jamming as a phase transition [450–452]. Such models are quantitatively consistent with in vitro observations of the steep drop in both velocity and run-length when the density of kinesin-1 motors on a microtubule reaches a critical value [453]. Because traffic jams depend both on total motor density and accumulation at microtubule ends, the moderate processivity and high end detachment rates of kinesin have been hypothesized to be advantageous for overall cellular transport [454]. Interestingly, for cargo that can bind motors reversibly, increased free motor density can actually give rise to longer run-lengths [455], possibly due to the cargo's ability to associate with more motors to bypass localized traffic jams or effectively surf along densely packed neighboring motors [442].

The motion of a motor-driven cargo along a single microtubule is determined by a complex interaction between the complement and regulation of motors attached to the cargo, the distribution of roadblocks and traffic jams along the microtubule, and the presence of cytoplasmic obstacles encountered by the cargo. We next proceed to consider how cargo distribution on a cellular scale is governed by a combination of limited-processivity runs interspersed with passive periods.

Cellular cargos engaged in long-range transport often undergo periods of processive runs interspersed with pauses of varying duration [19, 26, 382, 393–395]. The pauses can be very long, as is the case for axonal mitochondria that have been observed to switch from a motile to a long-lived stationary state [165]. They can also be transient, associated with maneuvering around an obstacle [2, 446], tug-of-war between opposing molecular motors [382, 456], or dissociation from the microtubule or hitchhiking carrier [26, 457]. During such pauses the cargo can remain stationary, tethered to the microtubule itself or to nearby filaments of the actin cytoskeleton [165, 458, 459]. Alternatively, the cargo can be free to diffuse within the cytoplasm until the next run of processive motion [26, 457].

A simple mathematical model for transport consisting of interspersed periods of diffusive and processive motion is the one-dimensional 'halting creeper' (figure 14, inset) [25]. This model comprises one-dimensional particle motion, switching at fixed rate k_stop from processive motion with velocity ±v to pauses with diffusivity D and vice versa with rate k_start. Such a particle has a run length ℓ = vk_stop and is processive a fraction f = k_start/(k_start + k_stop) of its time. The 1D model is particularly relevant for particles within cellular regions that form highly extended tubules, such as fungal hyphae and neuronal axons.

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The transport range (length of domain explored) for a halting creeper particle transitions from a diffusion-dominated regime at short times, to a ballistic intermediate motion above a characteristic length scale x* which can be estimated by setting fPe(x*) > 1 [25]. At much longer length and time scales, when the particle has had the opportunity to sample repeatedly between the different modes, it again exhibits effectively diffusive transport (figure 14). Similar transitions, albeit on different time-scales, are also observed for the MSD of a particle engaged in multi-modal transport [409, 460].

The relative importance of diffusive versus processive motion thus depends on both the length scale of interest and the overall objective of transport. For instance, the uniform dispersion of an initially concentrated bolus of particles is optimized at intermediate values of the run length ℓ and of the active fraction f [25]. For particles that are only able to carry out their function in the passive state (e.g.: proteins that must be released from a vesicle), reaction kinetics are fastest at intermediate fractions of time in active motion [461]. Even for constantly active particles, when the domain is sufficiently long and f is sufficiently high, the search time for a single particle to hit a target is also optimized at intermediate run lengths, which preclude very long excursions in the wrong direction [462].

When the transport objective comprises efficient encounter of a target by the first in a uniform population of particles, the relevant length scale becomes the inverse of the particle spatial density, which tends to be on the order of 0.1–10 μm. For densities higher than 1/x*, target search is dominated by diffusive transport, whereas for lower densities motor-driven motion predominates. Interestingly, many organelles capable of motor-driven transport have been found to spend only a small fraction of time actually engaged in processive motion [26, 382, 392, 457]. These particles can have sufficiently high values of x* such that both diffusion and active transport contribute substantially to target search processes [25].

While the velocity v of processive motion is fairly constant (of order 1 μm s⁻¹), cells can regulate both the typical run length ℓ and the pause time t_pause = 1/k_start for transported particles. The pause time, in particular, can be reduced by tethering the particle to the microtubule track and thereby increasing the rate at which it can re-engage with the machinery for motor-driven transport. Such tethering is particularly effective when the microtubules themselves are sparsely distributed and diffusion toward a microtubule becomes rate-limiting for initiating transport [25, 393]. Recent mechanical modeling of hitchhiking transport for fungal peroxisomes indicates that tethering to microtubules could enhance the rate of starting a hitchhiking run by up to an order of magnitude [393]. For directly motor-driven cargo, tethering and preventing dissociation from the microtubule has been proposed as a cooperativity mechanism for motors with opposing polarity [19]. In terms of transport efficiency, the enhanced starting rate for active motion due to tethering is balanced by reduced diffusive exploration during the paused state. For organelles that spend a small fraction of time engaged in processive motion, tethering is beneficial for transport only on length scales beyond ${L}_{\text{crit}}\approx {x}^{{\ast}}/{\left(1-{\hat{a}}^{2}\right)}^{2}$, where â is the ratio between the capture radius around a microtubule and the characteristic separation between parallel microtubules [25].

By tuning pause rates and durations, as well as the mobility state of a particle while paused, cells can thus regulate overall particle dispersion through an interplay of passive and processively moving transport modes.

The intracellular distribution of cargos and their efficiency at reaching cellular regions can be controlled at several levels. As discussed in section 4.2.2, biochemical modification or binding of signaling molecules to motor-proteins, adaptors, linkers, and cytoskeletal tracks can regulate the processivity and directional bias of cargo moving along a single microtubule. However, models of transport that rely on uniform constant-rate processes at the single-cargo level tend to be insufficient to reproduce the complex behavior of motor-driven cargos in vivo [19, 415, 416]. Some of this complexity may be due to spatially or temporally heterogeneous regulation, with gradients in signaling molecules responsible for modulating transport parameters in different regions of the cell [185, 188, 435]. However, an additional key source of spatial heterogeneity is the organization of the cytoskeletal highways themselves. This organization both determines and is set by cell shape and polarity, allowing for a close coupling between cellular function, morphology, and transport logistics [364]. In some systems, incorporating the explicit distribution of cytoskeletal filaments has been shown to be sufficient to explain observed transport behavior while maintainining spatially uniform cargo unbinding rates [469–471].

The two types of cytoskeletal filaments serving as transport highways exhibit very different organizations within the cell. Actin filaments tend to form branched networks of varying densities. In mammalian cells, dense actin networks are usually restricted to a cortical layer (∼100 nm thick) beneath the cell membrane [472]. Away from the leading edge of migrating cells, these cortical actin filaments tend to be isotropic, without a defined polarity [473]. Consequently, transport within the actin network tends to appear characteristically diffusive, even when driven by motor proteins [366, 461]. The effective diffusivity of particles moving within the actin network is thought to be regulated in different cellular states by altering the switching probability at each filament intersection, thereby controlling the processive run-length of the cargo [366].

By contrast, the microtubule cytoskeleton can form a variety of structures with different degrees of polarity and spatial organization. Microtubules nucleate at discrete sites termed MTOCs, which anchor their minus ends while allowing plus ends to grow outward. The best-studied MTOC in animal cells is the centrosome, which is located near the nucleus, and nucleates an aster-like structure of microtubules extending their plus ends toward the cell periphery [463] (figure 15(b)). At the periphery, microtubules can penetrate the cortical actin network, allowing cargos to switch from long-range transport on microtubules to short range motion on actin filaments [474], in a manner dependent on the complement of attached motors [475]. A number of non-centrosomal microtubule-organizing structures have also been identified, allowing for anchoring of minus ends in many different cellular regions, and giving rise to microtubule networks with varying polarity and orientational alignment [463] (figure 15). Some MTOCs are associated with the Golgi body and its outposts, allowing for direct delivery of dynein-driven vesicles carrying secretory cargo from the ER to the Golgi [476].

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In certain cell types, including Drosophila [477] and Xenopus [478] oocytes as well as epithelial cells [479], microtubule nucleation is localized at the cell cortex (figures 15(c)–(e)). While fully polarized epithelial cells can establish unidirectional microtubule structures (figure 15(e)), oocytes tend to exhibit largely disordered cytoskeletal organization [477]. Nevertheless, a statistical bias in microtubule orientation can be sufficient to enable robust localization of cellular components [358, 470, 471, 480]. For Drosophila oocytes in particular, a gradient of microtubule nucleation densities at the cell periphery was shown to be sufficient to establish a structured velocity field for motor-driven motion throughout the ooplasm, when averaged over many realizations of a cytoskeleton that turns over on minute time-scales [470]. The resulting orientational bias allows kinesin-driven mRNA molecules to accumulate at the posterior pole despite executing many rapid runs in all directions [481]. Simulation studies incorporating the biased orientation field accurately reproduce both this posterior localization and the more complex splitting behavior of dynein-driven mRNAs, whose ultimate localization depends on the point of injection [470, 482].

Elongated cellular regions, such as neuronal projections or fungal hyphae, generally exhibit arrays of parallel microtubules, arranged into polarized bundles [483]. Microtubules in neuronal axons are uniformly oriented, with their plus ends pointing to the distal end of the projection [484]. In dendrites, the orientation can be uniform with minus end outwards (in Drosophila and Caenorhabditis elegans neurons) [484] or mixed with plus ends in both directions (in vertebrate neurons) [485]. The ability of cargos to be transported selectively to dendrites or axons is thought to rely on varying recruitment of motor subtypes [486] together with post-translational modifications of the microtubule tracks [483].

The parallel architecture of microtubules in these cellular projections is conducive to modeling studies that treat the system as essentially one-dimensional, representing the density of microtubules, cargos, and motors as mean-field distributions along the axis of the projection. For example, a model of dynein-driven dendritic transport showed that microtubule arrays of mixed polarity resulted in slower delivery of cargo to the dendrite tip but more efficient establishment of a uniform distribution of cargos within the dendrites [486]. Modeling of early endosome transport in fungal hyphae demonstrated that spatially uniform rates of motor switching and microtubule nucleation are sufficient to reproduce experimentally observed accumulation of endosomes in different hyphal regions in response to dynein and kinesin-3 motor mutations [469].

Additional effects beyond a purely one-dimensional system arise when considering the radial spacing of microtubules within a cylindrical cellular projection. Because motor-driven organelle transport can only be initiated when the organelle passes close to a microtubule track, the cross-sectional movement of organelles can play an important role in their dispersion. Modeling of 3D particle dynamics has shown, for instance, that tethering of hitchhiking peroxisomes to microtubule tracks is expected to greatly increase their overall rate of transport, particularly when there are very few parallel microtubules in the cellular region [393]. Cylindrical models with explicit microtubule arrangements form a natural transition from one-dimensional models to local regions of fully 3D systems that are lacking in microtubule intersections. For example, the asymmetric densities of parallel microtubules observed in Drosophila cell spindles can be incorporated into a 1D transport model that explains the uneven distribution of endosomes between daughter cells [487]. Other modeling efforts have shown that random spacing of locally parallel microtubules leads to a higher long-range effective diffusivity of motor-driven particles than does purely uniform spacing [471].

In many animal cell types, microtubules form 3D networks with frequent intersections between individual filaments [2, 356]. These intersections serve as both obstacles for cargo moving along a microtubule (section 4.2.3) and as an opportunity to alter the direction of motion. The probability of switching tracks at a microtubule intersection is dependent on the 3D spacing and orientation of the intersecting microtubules [2, 447], as well as the cargo size [445] and complement of attached motors [448]. Live-cell tracking studies indicate that most cargos tend to preserve the anterograde or retrograde polarity of their motion upon passing microtubule intersections [2], an effect which may arise from the radially polarized organization of the microtubule network.

As with one-dimensional models, the motion of motor-driven particles over cytoskeletal networks is generally assumed to consist of stochastic switching between processive runs along filaments and slow passive phases [461]. While the passive phases are generally treated as diffusive, they may also involve tethering to stationary structures [19, 458, 459]. Recent work in which the passive mode is treated as a continuous-time random walk with broadly distributed step times indicates that such intermittent motion would give rise to a characteristic distribution of first passage times to the cell periphery [488]. Namely, a peak of particles arriving at short times is expected, followed by a sustained long tail of sporadic particle arrivals—a biphasic pattern which has been observed for the exocytic release of insulin granules [489].

The density, spatial distribution, and polarity of cytoskeletal filaments in a 2D or 3D cellular region plays an important role in determining the overall transport of cargo. Denser networks of filaments allow cargos to spend more time in the actively moving phase. However, more dense networks also imply more frequent filament intersections and thus shorter processive runs. Simulations on randomly oriented 2D networks indicate that the mean first-passage time from a central nucleus to the cell periphery is largely determined by the total mass of cytoskeletal tracks, with faster transport at higher total filament content [490]. For the same total network densities, structures with a few long filaments tended to exhibit much greater variation in transit times than those with many short filaments, an effect arising from the presence of 'traps' where processively moving cargo is directed into a localized region of the network [490, 491]. The polarity of randomly scattered filaments plays an important role in determining transition times across the network, and reversing the polarity of a single filament can alter the first-passage times several-fold [491].

Spatially inhomogeneous network structures can also help optimize transport of intermittently motor-driven cargos. Regions of randomly oriented short filaments serve to locally enhance the effective particle diffusivity. Continuum models show that when such a region is placed closer to the center of a circular domain, the mean first-passage time of particles from the center to the domain boundary can be significantly decreased [490]. By contrast, when the goal of a transport system involves locating a specific narrow target on the periphery, then optimal search rates can be obtained by an ordered radial arrangement of polarized filaments in the cell bulk, coupled with a thin shell of random filaments near the periphery [474, 492]. In this case, cargo is delivered in a directed fashion to the peripheral layer, followed by effectively diffusive exploration of the boundary. Such a morphology is indeed observed in many cell types which maintain a radially polarized microtubule cytoskeleton originating at the centrosome near the nucleus and a thin largely disordered cortex of actin filaments that may contribute to localized cargo transport in peripheral or distal regions [368].

Motor-driven transport is a ubiquitous feature of eukaryotic cells. Its unique advantage lies in its ability to deliver and disperse cargo in an efficient and regulated manner that can be modified via a plethora of control parameters tuned for different cargos, cell types, and cellular states. The factors subject to cellular control include cytoskeletal organization, motor recruitment, processivity of individual motors, and cooperative interactions between motor teams. However, motor transport is limited in its maximum speed, has a high metabolic cost in ATP consumption, and requires additional complexity in the packaging of molecular components into vesicles or motor-driven complexes. An alternate mode of directed intracellular transport, the movement of particles by cytoplasmic flow, offers cells the opportunity to circumvent some of these challenges.

As discussed in section 2, flows of intracellular fluids generally lie in the regime of very low Reynold's numbers, where viscous forces dominate over inertia. In this 'Stokes flow' regime, fluid flows are laminar, particle velocities are proportional to applied forces, and flow patterns are established nearly instantaneously throughout the domain for any given pattern of applied stresses [21]. Such systems are subject to an effect which has been whimsically referred to as the 'scallop theorem', where time-reversing flows result in no net movement of the advective particles [22]. In essence, particles that are mixed by stirring in a low Reynold's number fluid can be un-mixed by repeating the same stirring motions in reverse [499, 510, 511]. As a result, simple oscillatory back-and-forth flows cannot, in and of themselves, result in particle transport. However, long-range transport can be achieved by the establishment of steady, persistent flow patterns (as for cytoplasmic streaming in plant cells [32, 512]) or by coordinated oscillations that propel material via peristalsis (as in the shuttle flows of slime molds [194, 197]).

The spatiotemporal distribution c(x, t) of particles subject to both diffusive motion and flow is described by the advection-diffusion-reaction equation [513]:

$$\begin{equation}\frac{\mathrm{d} \overrightarrow {c}}{\mathrm{d}t}= \overrightarrow {\nabla }\cdot \left(D \overrightarrow {\nabla }c\right)- \overrightarrow {\nabla }\cdot \left( \overrightarrow {v}c\right)+R\left(x,t\right),\end{equation} \tag{ 16 }$$

where D is the diffusivity, $\overrightarrow {v}$ the fluid flow field (which can vary over space and time), and R is a reaction term that describes sources or sinks that may arise from chemical reactions. This general equation can be leveraged to describe pattern formation and signal propagation in a variety of cellular systems with cytoplasmic flow [28, 506]. The importance of flow versus diffusion over a length scale L is characterized by the Péclet number Pe(L) [32], which is defined generally for directed transport processes (equation (2)).

A large Péclet number (Pe ≫ 1) indicates advection-dominated transport. For non-stationary flows, the length scale can be replaced by L = vτ, where τ is the persistence time of the flow pattern. Cellular transport systems where advection is believed to play an important biological role have Péclet numbers in the range Pe ≈ 2–1000, as summarized in table 1.

Several distinct mechanisms are capable of generating intracellular flows. The first mechanism relies on the contraction of actin filament networks by myosin motors. Large-scale flow patterns have been observed in reconstituted in vitro active gel systems with actin turnover and myosin activity [517, 518]. Waves of actomyosin contraction are responsible for the peristaltic shuttle flows in slime mold hyphae [197, 519], as well as flows that drive spindle positioning in mammalian oocytes [520] and nuclei dispersion in Drosophila embryos [508]. Myosin-driven contraction at the cell rear also drives flow toward the leading edge in migrating keratocytes [521] and neutrophil cells [522]. These flows can be regulated by gradients in the distribution of myosin motors or of signaling molecules that trigger myosin activation. When the molecules regulating contraction are driven by the flow itself, precise patterning of flows and molecular distributions can be established across the entire cell [28, 508, 523, 524]. Example flow patterns generated by actomyosin contraction are shown in figures 16(a)–(c).

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Large-scale contraction of the actomyosin network is often associated with deformation of the cell shape during migration [496, 522, 525], division [526, 527], and development [498, 528]. In many cases, however, cell shape dynamics are driven primarily by leading edge extension through directed polymerization of the actin cytoskeleton [529–531], as in the migrating neutrophil-like cell in figure 16(d). Growing cells, such as fungal hyphae, may also harness gradients in osmotic or turgor pressure to drive flow toward extending tips [497, 532] (figure 16(e)). Regardless of its origin, deformation of the cell boundary gives rise to cytoplasmic flows that can contribute to intracellular mixing [3] or overall translation of the cytoplasm [514].

An additional major source of flow is hydrodynamic entrainment by motor-driven cargo. Long-range, persistent flows are particularly prominent in plant cells (figures 16(f) and (g)), where myosin motors carry a variety of organelles along bundled actin filaments organized around the cell periphery [29, 499, 533, 534]. The motion of these organelles entrains a thick layer of cytoplasmic fluid, resulting in streaming flows that can reach 100 μm s⁻¹ [534]. In animal cells, entrainment-driven flows tend to be slower and more spatially heterogeneous. In Drosophila oocytes, for instance, kinesin-bound cargos are responsible for slow, apparently random flows (25 nm s⁻¹) and rapid, coordinated streaming (300 nm s⁻¹) during different stages of oogenesis [466] (figures 16(h) and (i)). Seemingly random flow patterns in the early oocyte tend to be spatially correlated on the few-micron scale (figure 16(h)), likely due to the underlying organization of the microtubule cytoskeleton [86, 507, 538] (see figures 15(c) and (c')).

In other systems, where cellular-scale flows are not directly evident, the bidirectional motion of motor-driven cargos may nevertheless give rise to very short-range entrainment events for nearby tracer particles [107]. When the cargo motion is slightly biased toward one direction, an overall slow flow of passive cytoplasmic contents will arise. For example, a bias toward anterograde cargo motion in growing fungal tips has been suggested to give rise to a very slow directed polar drift (0.5 nm s⁻¹) that leads to organelle accumulation when other active transport mechanisms are removed [26].

The variety of spatiotemporal flow patterns generated by different cellular mechanisms contributes to the distribution and dispersion of intracellular particles ranging from small nutrient molecules to proteins and organelles. Unlike diffusion, flows can drive the motion of even very large particles. Unlike motor-driven active transport, they affect all particles passing a particular region, without the level of regulation derived from specific adaptors coupling motors to cargos. We proceed to consider the functional consequences of various flow patterns on both directed localization of cellular components and overall mixing of cell contents.

Stable, persistent cytoplasmic flow provides a mechanism for directed transport of cellular components, allowing the establishment of intracellular gradients and the localized positioning of organelles. In mammalian oocytes, cytoplasmic flow drives the placement of the meiotic spindle near the cortical cap [520]. In C. elegans zygotes, flows with Pe ≈ 3 contribute to the anterior accumulation of PAR proteins [506, 535]. Directional advective transport also contributes to delivering cytoplasmic contents that drive cellular growth in a variety of systems, including the developing axon [536], slime mold plasmodium [194], fungal hypha [497], and elongated plant and algal cells [32, 512].

The simplest model for localization and gradient-formation by advection consists of a one-dimensional domain of length L with reflecting boundary conditions and a steady unidirectional flow of velocity v. The steady-state distribution of a particle with diffusivity D is then given by solution of equation (16) as

$$\begin{equation}c\left(x\right)=\frac{\overline{c}\enspace \text{Pe}\enspace {\mathrm{e}}^{\text{Pe}\cdot \left(x/L\right)}}{{\mathrm{e}}^{\text{Pe}}-1}\end{equation} \tag{ 17 }$$

where $\overline{c}$ is the average density and Pe is the Péclet number over the domain. High Péclet numbers lead to sharp accumulation of density at the domain boundary, while lower values result in a more uniform distribution (figure 17). Because diffusivity generally scales with particle size, larger particles develop sharper gradients under a given flow—an effect that has been used to estimate flow velocities in the leading edge of crawling keratocytes [521].

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Gradients can be further enhanced by a polarized distribution of molecules capable of binding the particle of interest (figure 17). Weak binding, along with directed flow, can combine to segregate a molecule into a specific cellular region, while allowing for rapid equilibration within that region. An analogous mechanism has recently been shown to underlie the accumulation of proteins in the outer segment of mammalian photoreceptor cells [537]. It should be noted that the distributions described by equation (17) and its generalizations do not require that v represent fluid flow specifically. Any kind of directed transport process that moves all relevant particles passing a particular point in space with the same velocity can supply the advective drift v. This could include, for instance, the IFT trains that transport proteins into primary cilia [142, 143]. By interacting only with certain specific proteins, such forms of directed transport allow for more precise control over the patterning and accumulation of intracellular particles.

Conservation of mass implies that when advective flow delivers cytoplasmic contents to specific cellular regions, the fluid itself must either recirculate or deform the cell contour. In growing or migrating cells, expansion of protrusions provides a reservoir for newly arriving cytoplasm (figures 16(a) and (d)). Other transport systems rely on fountain flow patterns (figures 16(c) and (f)) that cycle the incoming fluid with peripheral flow in the reverse direction from flow along the central axis. In these flow patterns, local binding or rapid removal via metabolism or exocytosis is needed to prevent newly delivered molecules from being flushed back by the recirculating flow [516]. Yet another pattern of advective delivery is seen in some fungal hyphae, where flows pass between cellular regions separated by septa with a narrow central pore (figure 16(e)). The focusing of flow through the pore leads to the formation of circular eddies on the upstream side of the septum, which can serve as a subcellular compartment. These compartments locally entrap nuclei that proceed to differentiate to a transcriptional program which differs from other nuclei in the same cytoplasm [515]. Furthermore, the flow-driven accumulation of vesicles at the septa has been hypothesized to contribute to hyphal branch formation [191].

The entrainment of cytoplasm by motor-driven cargo also raises the problem of fluid cycling when the cargo approaches the end of a cellular region. Modeling studies indicate that the recirculatory flow engendered by this entrainment may counteract directed transport, washing unbound cargo and other passive particles out of the target zone [538]. The resultant coupling between motor-driven motion and advective flow implies that disordered, weakly directional cytoskeletal networks may in fact lead to more optimal local accumulation of particles [538].

Many cellular advective transport systems rely on relatively stationary flow patterns that persist over sufficient time periods to enable particle delivery across the cell. However, important counterexamples exist, where large-scale directed movement of cytoplasmic contents is achieved through coordinated time-varying flows. A particularly well-studied example is the peristaltic shuttle flow observed in slime molds, both in their migrating ameboid [496] and their hyphal network [194] state. These flows are generated by directionally propagating contraction fronts that are thought to be self-organizing via a signaling molecule that both amplifies contractions and is advected by the flow itself [28]. In general, peristaltic flows require an organized spatial gradient of contraction phases, allowing for overall directed transport of fluid contents [539]. In tubular network structures, advective transport is optimized when the wavelength of the peristaltic wave is comparable to the network size, consistent with the observed phase correlation patterns in P. polycephalum hyphae [194]. An alternate example of cytoplasmic transport by oscillatory flows has recently been observed in multinucleate Drosophila embryos, where vortex-like flow patterns (figure 16(c)) oscillate in coordination with the cell cycle. These flows are able to drive the separation of nuclei originally clustered near the embryo center to well-spaced positions along the anterior–posterior axis [508].

In addition to targeted delivery and patterning, cytoplasmic flows can also drive more efficient mixing of cellular components. Mixing in the world of low Reynold's number fluids relies on two distinct physical effects: Taylor dispersion (the smearing out of concentration gradients by diffusion) [541, 542] and Lagrangian stirring (the chaotic motion of particles driven by a spatially heterogeneous, unsteady, non-reversing flow) [543, 544]. Taylor dispersion arises from spatially varying rates of flow, which give rise to gradients in particle densities, resulting in an effectively higher diffusivity of particles across the streamlines (figure 18(a)). For steady Poiseuille flow in a tube [545], the effective diffusivity along the cross-section of the tube is given by

$$\begin{equation}{D}_{\text{eff}}=D\left(1+{\text{Pe}}^{2}/48\right),\end{equation} \tag{ 18 }$$

where Pe refers to the Péclet number (equation (2)) computed for the average velocity in the tube over the length-scale of the tube radius. For the rapid contractile flows in P. polycephalum (velocity ≈ 0.1 mm s⁻¹, radius ≈ 50 μm, Pe ≈ 50), the effective dispersion of small molecules (defined as ${\left\langle 1/{D}_{\text{eff}}\right\rangle }^{-1}$) is increased by up to 7-fold [195].

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Several cellular systems with more complicated flow patterns have also been hypothesized to enhance diffusive transport through the flow-induced formation of steep gradients. In late-stage Drosophila oocytes, streaming flows exhibit faster velocities toward the cortex (figure 16(i)), leading to cytoplasmic shear gradients that may contribute to mixing [466]. In the long cylindrical cells of characean algae, high shear rates result from rapid spiral cytoplasmic streaming [29, 32, 500]. These flows are expected to give rise to radial concentration gradients that augment diffusive entry of nutrients into the cell (figure 18(b)). Interestingly, the geometry of flow patterns can be used to tune the gradient steepness and hence the rate of mixing or diffusive uptake. Internodal cells of the algae Nitella axillaris alter the wavelength of their spiral flows as the cell grows, with a maximum in both diffusive uptake and growth rate arising at a specific cell length [29]. Foraging P. polycephalum slime molds prune their network structure to increase flow speeds in a few large central tubules, increasing particle dispersion [195].

In addition to Taylor dispersion, Lagrangian stirring resulting from unsteady fluid flow patterns also contributes to mixing in cellular systems, even for particles whose diffusion is negligible. Stirring is often described by quantifying the extent to which a given region of fluid stretches and folds under the flow (figure 18(c)), increasing the length of its boundary with the surrounding fluid [544]. These boundaries mark regions of high gradients, which can then be smoothed by diffusion. Stirring thus acts together with Taylor dispersion to mix the system across different length scales. Lagrangian stirring arises from the fact that even a relatively simple laminar flow pattern for a low Reynold's number fluid can nevertheless lead to highly complex (chaotic) trajectories of individual particles or fluid elements [543, 546, 547]. Chaotic trajectories are characterized by positive Lyapunov exponents [544], which quantify the exponential divergence of paths for two initially close particles carried by the fluid. Although a steady flow field can yield such diverging trajectories in three dimensions, time-varying flow patterns are needed to generate chaotic stirring in 2D fluids [544, 547]. An example of diverging particle trajectories due to unsteady flow is seen in the spreading of nuclei along the anterior–posterior axis of Drosophila embryos (figure 18(d)) [508].

In practice, extensive and efficient stirring can be achieved by unsteady flows that are not time-reversing. In such systems, the Péclet number associated with instantaneous flow velocities does not adequately describe the overall stirring behavior, since partial flow reversals tend to drive particles back toward their starting points. Instead, one can characterize the effect of flow on mixing by defining an 'effective Péclet number' on any given time-scale, as the overall displacement of a tracer particle driven by flow alone versus the diffusive displacement over the same time period [3]. Starfish oocytes serve an example of a cellular system with rapid back and forth flows (figure 16(b)) but no significant overall displacement of large cytoplasmic particles [498]. More complex dynamically evolving flow patterns are observed in the cytoplasm of crawling cells executing amoeboid-like deformations [3, 496, 548]. Numerical simulations indicate that the flows arising from deformation of neutrophil-like migrating cells (figure 16(d)) are sufficient to substantially enhance the mixing of lysosome-like organelles in the cytoplasm (Pe_eff ≈ 11 over 30 s timescales) [3]. In late-stage Drosophila oocytes, dynamic buckling of microtubule tracks due to drag forces on motor-driven cargo is thought to give rise to local time-variation in the overall flow pattern [466, 549]. The resulting unsteady flows (figures 16(h) and (i)) have been shown to homogenize the distribution of initially concentrated yolk granules within the cytoplasm [466, 550].

Locally oscillating flows can efficiently drive dispersion when the particles are confined in a domain of complex geometry and the overall spatial pattern of flows is stochastic. A biologically relevant example is the luminal flow generated by random contractions in a tubular network, as in slime-mold hyphae [195]. In order for such flows to contribute substantially to mixing, they must be rapid enough and persistent enough to enable individual particles to transition between nodes before the contraction re-opens, reversing the flow. Once a particle reaches a network junction, flow splitting and small time delays in flow reversal at adjacent edges ensure that the particle does not get restored to its initial position, thereby promoting mixing through Lagrangian stirring [551].

On a smaller scale, flows arising from uncoordinated tubular contractions have recently been hypothesized to drive node-to-node transport of proteins in the mammalian ER network [59]. Processive particle velocities on the order of 20 μm s⁻¹, over time scales of 30 ms, have been measured for individual proteins tracked in ER tubules, which exhibit a luminal diffusivity in the nodes of about 0.5 μm² s⁻¹ [59]. These estimates yield a Péclet number of Pe ≈ 20 and allow for individual processive trajectories to cover a distance comparable to the typical edge length in an ER network (∼1.5 μm).

An additional role for stochastic intracellular flows in driving cytoplasmic mixing is through uncoordinated entrainment by motor-driven cargos. Such entrainment events result in short-range runs, leading to an enhanced 'active diffusion' driven by bidirectionally moving cargos. Localized entrainment has been hypothesized to account for the 'slow component' of axonal transport [107] and the kinesin-dependent 'active diffusion' of peroxisomes in fungal hyphae [26], although direct evidence of their importance in cellular transport is still lacking. In a tubular system, each entrainment event from the passage of a single organelle should yield a finite short displacement (ℓ) of a tracer particle of length comparable to the organelle size [107] (figure 18(e)). If organelles pass near the tracer at a frequency k_pass, the effective diffusion coefficient for the tracer is then given by

$$\begin{equation}{D}_{\text{eff}}=D+{\ell }^{2}{k}_{\text{pass}}\end{equation} \tag{ 19 }$$

where D is the tracer diffusivity in the absence of active motion. In fungal hyphae, knocking out an endosomal motor adaptor results in a decrease in the diffusivity of passive peroxisome organelles by ΔD ≈ 0.01 μm² [26]. This effect is comparable to the predicted contribution due to entrainment by passing endosomes, at a frequency of k_pass ≈ 1/s [392], in accordance with equation (19).

Flow of cytoplasmic fluids thus constitutes a versatile mechanism for transport across a broad range of length scales. Coordinated patterns of flow can result in the directed delivery of bulk cytoplasmic contents at speeds far higher than those reached by motor-driven transport. Furthermore, flows contribute to mixing of cytoplasmic contents through the formation of gradients smoothed by Taylor dispersion, through Lagrangian stirring, and potentially through the generation of effectively diffusive active motion via stochastic local entrainment events.