Drag of the Cytosol as a Transport Mechanism in Neurons

Abstract

Axonal transport is typically divided into two components, which can be distinguished by their mean velocity. The fast component includes steady trafficking of different organelles and vesicles actively transported by motor proteins. The slow component comprises nonmembranous materials that undergo infrequent bidirectional motion. The underlying mechanism of slow axonal transport has been under debate during the past three decades. We propose a simple displacement mechanism that may be central for the distribution of molecules not carried by vesicles. It relies on the cytoplasmic drag induced by organelle movement and readily accounts for key experimental observations pertaining to slow-component transport. The induced cytoplasmic drag is predicted to depend mainly on the distribution of microtubules in the axon and the organelle transport rate.

Introduction

Neurons are highly polarized cells with axons capable of extending long distances. Individual axons are microscopic in diameter (typically of the order of 1 μm), whereas their lengths may span as much as one meter in humans (e.g., axons in the sciatic nerve and spinal cord) and even more in large animals such as giraffes or whales. Since axons are cellular structures with a unique morphology, the morphogenesis, integrity, and function of neurons are particularly dependent on active intracellular transport, also called axonal transport (1,2).

There are three known major constituents of the neuronal cytoskeleton: microtubules, microfilaments, and neurofilaments (2). In the axon and dendrites, microtubules and neurofilaments are the major longitudinal cytoskeletal filaments. In the synaptic regions, such as the presynaptic terminals and postsynaptic spines, microfilaments and microtubules form the major cytoskeletal architecture (3). There are also microfilaments along the axon, which mainly provide a scaffold beneath the axonal plasma membrane and around microtubules (4,5).

The lines of transport, communication, and control in the cell are provided by microtubules. These dynamic polymers create a network of tracks for motor proteins that carry various vesicles and organelles (referred to hereafter as cargoes), with diameters of the order of 10–100 nm (6). Microtubules appear as bundles of tubes dispersed across the axon and are aligned mostly along the axon axis to allow long-range transport (see, e.g., Fig. 8-1 in Siegel (2)). In comparison to the two other intracellular cytoskeletal filaments, microtubules are the most rigid, thus ensuring mechanical stability under the forces of large cargo movement (7).

Axonal transport plays a few major roles. 1), It is responsible for bidirectional molecular signaling over long distances. 2), It provides essential cellular organelles and materials from the cell body to the entire axon (anterograde transport by the kinesin motor protein). This supply of newly synthesized proteins and lipids to the distal synapse maintains axonal activity. 3), It causes molecules destined for degradation to return from the periphery to the lysosomes in the soma (retrograde transport by the dynein motor protein) (8,9). The modes of transport through the axon are typically divided into two categories: a fast component and a slow component (1). The components are distinct in their mean velocity, as measured in studies of radioactively labeled proteins (10–15).

The fast component is a directed movement of membranous cargoes. Small cargoes move continuously in one direction and maintain almost uniform velocity that ranges between 0.4 and 3 μm/s, whereas larger cargoes exhibit irregular saltatory motion (16–21). The fast transport is carried primarily by molecular motor proteins that bind directly to the cargo surface (22–24). The directionality of the cargo depends on the motor type and the polarity of the microtubule tracks (1).

The slow component (SC), which is the focus of this work, is usually divided into two subgroups with distinct overall rates: SCa and SCb (1). SCa transports cytoskeletal proteins, whereas SCb transports hundreds of different proteins that are important for synaptic and axonal maintenance, including metabolic enzymes, chaperons, presynaptic proteins, and cytoskeletal proteins (see Lasek and colleagues (25,26), Baitinger and Willard (27), Nixon et al. (28), Yuan et al. (29), Bourke et al. (30), and Roy et al. (31) and references therein). Key experimental observations concerning both SCa and SCb are as follows (31–35). 1), Transported soluble molecules spend most of their time pausing, and their movement appears bidirectional. 2), They are not attached to vesicles but homogeneously distributed throughout the entire cytoplasm. However, their transport was found to depend on motor proteins (35). 3), They move at broadly distributed velocities, ranging from close to zero to the velocity of the fast component, i.e., a few μm/s. 4), Slow-component proteins were even observed to move side by side with fast-component cargoes (34).

In contrast to the transport mechanism of the fast component, that of the SC is still under debate (36). Blum and Reed (37) suggested a set of interactions between soluble proteins and motor proteins. When bound to the motor, the soluble proteins move, and when not bound, they are stationary. Additional stochastic models have also been suggested (38–41) in which soluble proteins bound to motors move in the cytoplasm at different discrete velocity states and in which the transition from motion to halt is a probabilistic one. Scott et al. (35) proposed that soluble proteins move in the cytoplasm through both passive diffusion and a direct interaction with hypothetical mobile units. In that model, the assembly and disassembly from the mobile units is probabilistic. Miller and Heidmann (36) offered a qualitative description according to which soluble protein dynamics is governed by several mechanisms, including diffusion from high to low concentrations, vesicles dispersing soluble proteins as they travel, and displacement induced by the elongation of axons.

In this work, we propose an alternative, simpler, mechanism for SC axonal transport. It relies on the indirect hydrodynamic interaction (flow-induced drag) that exists between soluble particles and actively transported cargoes along the axon. We show, without further conjecture, that this simple mechanism is sufficient to account for the most relevant experimental observations mentioned above. An illustrative metaphor for the suggested mechanism is that of log driving, i.e., the transportation of logs down a river. However, in the case of intracellular drag, the river flows intermittently, being driven by anterogradely and retrogradely moving vesicles, rather than by a steady external field (gravity).

Earlier theoretical works addressing cytoplasmic flow along the elongated cells of Characean algae (42–45) are worthy of mention here. Although those studies also considered flows induced by actively moving organelles, their flow scenarios differ qualitatively in their geometry from that considered here.

The outline of the article is as follows. In the next section, we describe the model and its assumptions. Technical methods used to analyze the model are explained in the Methods section. In the Results section, we provide detailed data from the analysis in different scenarios. In the Discussion section, we discuss specific implications of the analysis for actual axonal transport, as well as further implications.

Model

The system under consideration is schematically shown in Fig. 1.

Figure 1.

(a) Diagram highlighting key features in axonal cross section. Black dots represent microtubules going through the plane of the figure and distributed uniformly over the cross section. One cargo is attached to a microtubule at the upper right corner of the axon. For more details, see Bray and Bunge (46). (b) Schematic diagram of the model. Spheres of radius a move at constant velocity along the axial direction of a cylindrical tube of radius R and length L.

We model the axon as a rigid, fluid-filled, cylindrical tube of length L and radius R. The problem is treated using cylindrical coordinates r = (ρ, φ, z), where the tube axis lies along the $\hat{z}$ direction. The cargoes are modeled as rigid spheres of radius a moving inside and along the tube. Effects related to shape changes of both the axon and the cargoes, as well as the internal dynamics inside the cargoes, are thus neglected. We assume that each cargo moves at its own constant velocity, U_i, parallel to the axis of the tube (21). The tube is filled with a viscous Newtonian fluid at zero Reynolds number (47) having viscosity η. This may seem a crude oversimplification, given the complexity of the cytoplasm. However, based on our results, the detailed response of the fluid should not have a strong effect on the long-range transport through the tube. We return to this issue in the Discussion. The hydrodynamics of particles moving through fluid-filled tubes have been widely studied (see, e.g., Brenner and colleagues (47–49), Liron and Shahar (50), and Blake (51)). Here, we focus on the fluid displacement in the Lagrangian framework, i.e., the displacement of fluid particles resulting from the active motion of the cargoes. We consider the transport of the SC as a passive entrainment of soluble proteins by these active flows. This assumption relies on the fact that the proteins (a few nanometers in diameter) are much smaller than the cargoes (with radius a on the order of 10–100 nm).

Flow problems in such confined geometries are highly dependent on boundary conditions (52). For example, when a sphere moves an infinite distance in an unbounded quiescent fluid, the net displacement of any fluid particle is infinite (53). This idealized result arises from the slow 1/r decay of fluid velocity with distance r away from the moving sphere. The opposite example is of a sphere moving in a closed tube with no-slip boundary conditions. The fluid velocity induced by the sphere decays exponentially with distance (50), resulting in a finite displacement of fluid particles.

A key question is the extent of solvent permeation through the boundaries along the axon. It is known that fluid diffuses through the membrane, e.g., via aquaporin channels (54). However, how appreciable such permeation is as a result of cargo movement remains unclear. Hence, we treat two simplified limits for the boundary conditions to be taken at the tube edges. 1), For the case of negligible permeation, we model the axon as a closed tube of a given length, L. 2), If permeation is significant, we replace the leaky axon with an open-ended impermeable tube of effective length L related to the membrane permeability, μ. The correspondence between these two systems is found by a simple analysis of a pressure-driven flow through a permeable tube, yielding a decay length of $L=\sqrt{R^3/\left(8\eta \mu \right)}$. (One can model the axon more accurately as an actual permeable tube, yet we prefer to defer such a detailed analysis to a future study.) We note that the flow patterns in these two limits are quite different. When a sphere moves in a closed tube in the axial direction, the fluid is quiescent far away from the sphere, whereas near the sphere, some fluid trajectories must go against the direction of motion due to mass conservation (50). By contrast, a similar sphere moving through an open-ended tube creates a Poiseuille flow far from the sphere in the direction of motion (55).

An additional property that may influence the fluid behavior is the friction at the tube walls. For example, observations in the giant internodal cells of Characean algae showed that the fluid slides on the inner surface of the cell (56). Accordingly, slip boundary conditions were used in modeling that system (43). The appropriate boundary conditions applied by the axonal membrane and the thick protein layer adjacent to it are unknown. Therefore, we consider the entire range of possibilities, from the no-slip boundary condition through partial slip to full slip.

Methods

We numerically solve the Navier-Stokes equations for an incompressible fluid in the limit of zero Reynolds number (Stokes equations),

$$\begin{array}{c}-\nabla p+\eta {\nabla }^2\mathbf{v}=0\\ \nabla \times \mathbf{v}=0\end{array},$$ (1)

for a fluid confined inside a tube. In Eq. 1, v(ρ, φ, z) and p(ρ, φ, z) are the fluid velocity and pressure fields. The calculation is done using a finite-element software (COMSOL, version 4.2a). For a single sphere moving at velocity U, the calculation is done in the reference frame of the sphere, i.e., the sphere is at rest and the fluid flows past it with a far velocity of −U. Trajectories of fluid particles are subsequently calculated by integrating the velocity field and transforming back to the fluid reference frame. For several spheres, we employ a superposition approximation after confirming its validity for a number of spheres moving at the same velocity (see Results for details).

Equation 1 is Galilean-invariant and, therefore, the same in the fluid and sphere reference frames. The entire system, however, is not Galilean-invariant, which is reflected in different boundary conditions. We begin with the boundary conditions at the edge of the tube. For a closed tube the boundary conditions are zero fluid velocity, v(ρ, φ, 0) = v(ρ, φ, L) = 0, which in the reference frame of the sphere turn into conditions of constant velocity, $\mathbf{v}\left(\rho ,\phi ,0\right)=\mathbf{v}\left(\rho ,\phi ,L\right)=-U\hat{z}$. For an open-ended tube, the boundary conditions at the edges, in both reference frames, are zero pressure difference, p(ρ, φ, 0) = p(ρ, φ, L), such that the flow is caused solely by the motion of the sphere.

Concerning the boundary conditions along the tube walls, ρ = R, no-slip boundary conditions in the fluid frame, v(R, φ, z) = 0, turn into $\mathbf{v}\left(R,\phi ,z\right)=-U\hat{z}$ in the sphere frame. Full-slip boundary conditions, v_ρ(R, φ, z) = ∂_ρv_φ(R, φ, z) = ∂_ρv_z(R, φ, z) = 0, remain the same in both reference frames. Partial-slip boundary conditions in the fluid frame read, v_ρ(R, φ, z) = 0 and v_φ(R, φ, z) + ℓ∂_ρv_φ(R, φ, z) = v_z(R, φ, z) + ℓ∂_ρv_z(R, φ, z) = 0, where ℓ is the slip length. In the sphere frame, the right hand side of the last two conditions is changed from zero to $-U\hat{z}$. The actual application of the partial-slip conditions requires further consideration due to software limitations. We use the method presented in Wolff et al. (43): a thin fluid layer of width ϵ = 0.001R is added around the tube. The additional layer has a different viscosity, η_ϵ < η. The boundary conditions are no-slip at the external radius, R + ϵ, and full-slip at R. The relation between the slip length and these parameters is ℓ = ϵ(η/η_ϵ −1) + O(ϵ²) (43).

Results

Single sphere in a closed tube

Typical trajectories of fluid particles dragged by a moving sphere along the central axis (ρ = 0) of a closed tube with no-slip boundary conditions are depicted in Fig. 2 a. At a particular radial position, ρ = ρ₀, fluid particles move outward and circle back to their initial position (Fig. 2 a, inset). Fluid particles located near the center line (0 ≤ ρ < ρ₀) are dragged forward, whereas those at the periphery (ρ₀ < ρ < R) move backward. The value of ρ₀/R depends on a/R and L/R.

Figure 2.

(a) Typical trajectories of fluid particles due to the motion of a single cargo along the axis of a cylindrical tube. Backward particle trajectories are marked in red. (Inset) Enlargement of particle trajectories with positive, zero, and negative net displacement. (b) The displacement profile. (c) Dependence of the fluid displacement on a/R for four radial positions. Physical parameters are a/R = 0.1 (in a and b), L/R = 10. To see this figure in color, go online.

We define a net fluid displacement, d(r), as the total distance traversed by a fluid particle that started at position r, for given initial and final positions of the moving sphere, z_i and z_f. The inset of Fig. 2 a shows three trajectories with positive, zero, and negative net displacements. The displacement profile corresponding to the trajectories of Fig. 2 a is shown in Fig. 2 b. The displacement is zero at ρ = ρ₀, and for a no-slip tube, it becomes zero again at ρ = R.

We now discuss additional general properties of the net displacement profile. 1), It is independent of U and η. This can be shown by dimensional arguments and is also verified numerically. 2), If the initial and final positions of the sphere, and the initial position of the fluid particle, are all well separated, z_i ≪ z ≪ z_f, with |z_f − z_i| ≫ R, the displacement profile becomes independent of axial position d(r) = d(ρ,φ). This results from the fact that the fluid velocity field decays exponentially with z/R away from the moving sphere (50). In addition, in the case of azimuthal symmetry (the sphere moving along the central axis), the profile depends only on the radial position of the fluid particle, d(r) = d(ρ). 3), For the same reason, unless the fluid particle is arbitrarily close to the center line, the net displacement is achieved during a finite period of time. This local response in space and time, resulting from the confinement of the fluid, allows us to restrict the discussion hereafter to spheres that move along the entire tube (z_i = 0, z_f = L ≫ R). In this limit, d/R becomes a function of ρ/R and a/R only. 4), For sufficiently small values of a/R and a finite value of ρ/R, we recover the displacement profile in an unbounded fluid (53), with d proportional to a, as shown in Fig. 2 c. For small values of both ρ/R and a/R, the displacement profile has the form d = c₁a²/ρ−c₂aln(ρ/R) (53). 5), In a closed tube, due to mass conservation, the integral of the displacement profile vanishes, ∫ d (ρ, φ) ρdρdφ = 0.

The shape of d(ρ) (Fig. 2 b) may be divided into three regions. For $\rho \lesssim a$, the displacement is governed by the near field—the stage of motion when the sphere and fluid particle were close together. This behavior is similar to that in an unbounded fluid, with a displacement proportional to a times a function that diverges as ρ → 0. For $a\lesssim \rho \lesssim {\rho }_0$, the displacement is proportional to R, arising mainly from the stages outside the near field. Finally, for ${\rho }_0\lesssim \rho <R$, the (negative) displacement is governed by the opposite flow due to mass conservation.

We now consider the effect of partial slip at the tube walls. Fig. 3 a depicts the displacement profiles for several values of the slip length ℓ, compared against the profile of a no-slip (ℓ = 0) and full-slip (ℓ → ∞) tube.

Figure 3.

(a) Displacement profile for no-slip (black), full-slip (red), and partial-slip tubes with several ℓ values (blue). (b) Trajectory of a fluid particle, initially located at ρ_ℓ, for no-slip (black), full-slip (red), and several finite values of slip length, ℓ. Parameters: a/R = 0.3, L/R = 30. To see this figure in color, go online.

An important observation is that the effect of wall slip on the displacement profile is minor. In addition, we find a special radial position, ρ = ρ_ℓ, at which the fluid displacement is a constant independent of ℓ (see Fig. 3 a, inset). Closer to the axis, ρ < ρ_ℓ, the net displacement increases as ℓ is increased, whereas closer to the boundary, ρ > ρ_ℓ, the net displacement becomes more negative with the slip. We note that the insensitivity to ℓ is restricted to the total displacement. The detailed trajectories of fluid particles significantly depend on slip, as demonstrated in Fig. 3 b.

Another way to examine fluid drag is through the drift volume (57), defined as the volume enclosed between the initial and final positions of a marked fluid surface—here taken as the cross section of the tube. In the case of a closed tube, described here, the drift volume vanishes due to mass conservation, as explained above. We define the positive and negative contributions to the drift volume, V_pos and V_neg, which are the integrals of the displacement profile, d(ρ), for 0 ≤ ρ ≤ ρ₀ and ρ₀ ≤ ρ ≤ R, respectively, such that V_pos = |V_neg|. Fig. 4 shows the change in these partial volumes as the slip length, ℓ, is varied. Although they increase monotonically with ℓ, the increase is not very significant, reflecting again the insensitivity of the net fluid motion to the boundary conditions at the tube wall.

Figure 4.

Change in the partial volumes, V_pos = |V_neg|, as ℓ is varied. Parameters: a/R = 0.01, L/R = 6.

Many spheres in a closed tube

For a system with N simultaneously moving spheres in a closed tube, we introduce an approximate solution and verify it as valid if the separation between the spheres is larger than the tube radius R. Due to hydrodynamic screening, the fluid trajectories and the displacement profile can be calculated by superimposing the motions induced by each sphere separately. Furthermore, for trajectories and displacements occurring sufficiently far away from the tube walls, one gets good estimates by considering the much simpler situation, where each sphere moves along the symmetry axis of the tube. One subsequently shifts the resulting fluid motion back to the actual cross-sectional position of the sphere before superimposing the effects of all the spheres. This approximation does not satisfy the boundary conditions at the tube walls, yet the error is found to be small. For example, the total fluid displacement profile is calculated according to

$$d_N\left(x,y\right)\cong \sum _{i=1}^N{d}_1\left(x-x_i,y-y_i\right),$$ (2)

where (x_i, y_i) denotes the cross-sectional position of the ith sphere, and d₁ is the displacement induced by a single sphere moving along the central axis as obtained, for example, numerically.

We demonstrate the superposition approximation for 11 spheres arranged as shown in Fig. 5, and Fig. 6 compares the displacement profiles, d(ρ), as obtained from the full numerical solution and from the superposition approximation in Fig. 5. The approximation error is observable but small, on the order of a few percent.

Figure 5.

Geometry of 11 spheres arranged at five different initial radial positions. (a) Side view. (b) Cross-sectional view. Parameters: a/R = 0.1, L/R = 12.

Figure 6.

(a) Displacement profile of the system described in Fig. 5 as obtained numerically. (b) Approximate displacement profile as obtained from superposition. (c) Difference between the two displacement profiles. To see this figure in color, go online.

Open-ended tube

In an open-ended tube with zero pressure difference between the tube ends, the sphere movement generates in the entire tube a z- and φ-independent global flow, $\overline{v}\left(\rho \right)$, in addition to a fast-decaying local flow. The latter screened contribution is very similar to the flow produced by the sphere in a closed tube, as described in the previous section. This is demonstrated in Fig. 7.

Figure 7.

Fluid trajectories due to the motion of a single sphere in no-slip closed and open-ended tubes. Red trajectories are for an open-ended tube after subtraction of the global (Poiseuille) flow, and black trajectories are for a closed tube. (Inset) Enlargement of trajectories near ρ₀. Parameters: a/R = 0.1, L/R = 10. To see this figure in color, go online.

Consequently, the superposition approximation for N spheres in an open-ended tube takes the form

$$d_N\left(x,y\right)\cong \sum _{i=1}^N\left[d_1\left(x-x_i,y-y_i\right)+{\overline{v}}_i\left(\rho \right)t_i\right],$$ (3)

where d₁ is the single-sphere displacement profile discussed in the previous section, and t_i = L/U_i is the total time of motion of sphere i through the tube. Since ${\overline{v}}_i$ is proportional to U_i/L (58), the contributions of the global flows, ${\overline{v}}_i{t}_i$, add to the net displacement a constant, $\overline{d}\left(\rho \right)$, independent of U_i and L.

The contribution of the global flow to the net displacement in an open-ended tube, $\overline{d}$, makes it always larger than the net displacement in the corresponding closed tube. In addition, it makes the displacement sensitive to the boundary conditions on the tube wall. Both effects can be seen in Fig. 8. As the slip length is increased, the net displacement increases because of the reduced friction at the walls. Its profile flattens with increasing ℓ, as the global flow changes from parabolic to plug-shaped. For full-slip boundary conditions, the net displacement becomes proportional to the system size, L. Without the effect of the global flow, the curves shown in Fig. 8 become similar to those in Fig. 3 a.

Figure 8.

Displacement profiles induced by a single sphere in an open-ended tube for several slip lengths. The case of a no-slip closed tube is shown for reference. Parameters: a/R = 0.1, L/R = 10.

Discussion

Consequences for cytoplasmic drag in axons

Transport due to the motion of a single cargo

Let us estimate the mean displacement of a soluble molecule due to the motion of a single cargo. We assume a uniform distribution of ∼100 microtubules over an axonal cross section of ∼1 μm² (46,59). Each soluble molecule is therefore within a 50 nm distance of a microtubule. We consider the displacement of the molecule due to the motion of a cargo on the nearest microtubule only. Including cargoes moving farther away will only increase the mean displacement, and the estimate given below, therefore, is a lower bound. The mean displacement induced by a cargo moving along the nearest microtubule (i.e., averaged over 0 < ρ < 50 nm) is proportional to the cargo radius (see Results). Because of the slow radial decay, the proportionality factor is large (of order 10), as demonstrated in Fig. 9. The consequence is that a single cargo induces a ballistic motion of the fluid near the microtubule to a distance an order of magnitude larger than the cargo itself, which is of order 1 μm. Since the cargo moves at a typical velocity of 1 μm/s, this motion should last for a few seconds.

Figure 9.

Mean fluid displacement within a given distance from the microtubule (ρ_max/R = 0.05 or 0.1) as a function of a/R in a closed tube. Parameter: L/R = 15.

The rate of cargoes traveling along an axon was measured to be ∼1–100 cargoes/min in axons with a cross section of 1 μm² (60,61). This is a lower-bound estimation, since not all cargoes are visible in such experiments. We therefore consider a cargo rate of the order of 100 cargoes/min. Assuming that the cargoes are uniformly distributed among the microtubules, the rate of cargoes on a single microtubule, γ, is ∼1 cargo/min. Thus, on average, a soluble molecule is dragged ballistically for a few seconds every minute. In addition to the significant ballistic motion, the soluble molecule is expected to undergo smaller drags in both directions due to cargoes on more distant microtubules.

In summary, our lower-bound estimate yields an occasional micron-scale ballistic motion (stop-and-go motion) of a soluble molecule, lasting a few seconds. These results are consistent with those from experiments by Roy et al. (31,34). (Note that in those experiments, the analysis was deliberately restricted to proteins moving along large distances of >10 μm.) For illustration, Fig. 10 shows two such intermittent trajectories of a soluble molecule due to cargo motion that bear qualitative resemblance to trajectories reported in the Roy et al. studies (31,34).

Figure 10.

Two demonstrative trajectories of fluid particles along the axonal direction, as induced by the motions of three cargoes. Parameters: R = 1 μm, a/R = 0.1; cargo coordinates: (ρ₁, φ₁) = (0.5R, 0), (ρ₂, φ₂) = (0.1R, π) moving in the opposite direction, and (ρ₃, φ₃) = (0.4R, π); fluid particle coordinates: (ρ_a, φ_a) = (0.345R, 0), (ρ_b, φ_b) = (0.375R, π). To see this figure in color, go online.

Overall transport

We now turn to the overall transport of soluble molecules due to the motion of many cargoes. This transport is crucially dependent on the existence, or absence, of bias in the direction of cargo motion.

Let us begin by assuming no such bias. In such a case, fluid particles are transported randomly in both directions. Over a very long time (≫γ⁻¹, i.e., many minutes), their motion will become diffusive. The effective diffusion coefficient is D_eff ∼ γ〈d〉² ∼ 10⁻¹⁰ − 10⁻⁹ cm²/s. This is much smaller than the diffusion coefficient of water molecules inside cells (10⁻⁶ − 10⁻⁵ cm²/s (62)), and still smaller than that of proteins (∼10⁻⁸ cm²/s (63)) as measured in nerve cells. Thus, in the absence of bias, the cargo-induced motion considered here does not contribute appreciably to the overall transport over very long times. However, as discussed in the preceding section, it can occasionally make molecules move ballistically over micron-scale distances—a type of transport impossible to achieve through thermal diffusion.

On the other hand, a bias in the direction of cargo motion along the microtubules will create a net drift resulting in the accumulation of net displacement in the direction of the bias. Experimentally, a previous report indicates significant biases ranging from 10–30% for some cargoes, and up to complete unidirectional transport (e.g. for Macropinosomes or mitochondria in specific regions) (64). As an example, a 10% bias would give a mean fluid displacement of 0.1〈d〉/min, which is of the order of 0.1–1 mm/day for a closed tube. This is a conservative estimate; describing the axon as an open-ended tube with partial-slip boundary conditions would yield a larger net flow (note the effect in Fig. 8). These values are not inconsistent with the experimentally observed overall rate of SC particles: 0.1–1 mm/day for SCa and 1–10 mm/day for SCb (1).

Further implications

In this article, we propose that drag induced by active cargo motion may be a plausible mechanism contributing to the motion of SC proteins. The implications for axonal transport are discussed above. The essential conclusion is that fluid particles and soluble molecules should be dragged in a stop-and-go bidirectional motion along micron-scale distances during times on the order of seconds. This conclusion is in line with the known experimental facts concerning SC transport. Although additional mechanisms influencing the dynamics of SC proteins are not ruled out, the mechanism proposed here relies on simple physical effects without the need to add extra elements such as binding of the molecules to motor proteins (35).

Our analysis is based on two main assumptions. The first is that transported molecules are assumed to be well dissolved in the medium and therefore carried along with it as it flows. The theory does not hold for molecules that do not conform to this assumption, e.g., proteins that bind to intracellular structures.

The second major assumption is that the axonal medium is fluid, i.e., it should flow in response to the active motion of the cargo. Such flows were observed in response to the forced motion of magnetic particles (65). Evidently, the actual axoplasm is crowded, viscoelastic, and anisotropic (65). We have focused in this work on the simple question of whether the passive drag due to cargo motion is at all relevant or negligible as regards the SC of axonal transport. Within this modest goal, the orders of magnitude presented above should be valid for actual axons, based on the following arguments. 1), The complex forces exerted on the cargoes by molecular motors, and their resulting velocities—the well-documented fast component of axonal transport—are not addressed theoretically but are rather treated as external biological constraints imposed on our calculations of the drag of the much smaller dissolved proteins. 2), It is well established that confinement in a long tube must screen out hydrodynamic interactions over long distances—an effect that is directly related to the lack of momentum conservation in the confined fluid and remains valid regardless of the complexity of the fluid. 3), The environmental response to an object moving through the axon depends on size. For objects of molecular size, such as proteins, the medium’s response was found to be close to that of a simple solvent (66). 4), Concerning the long-distance flow induced by cargo motion, viscoelastic effects should set in at sufficiently large length- and time-scales. The only measurement of axoplasm viscoelasticity of which we are aware (65) gave comparable values for the storage and loss moduli over timescales of order 10³ s. Measurements in the cytoplasm of other cells (macrophages) (67) yielded a storage modulus of the same order as, or one order of magnitude larger than, the loss modulus for timescales of order 10 s. Related materials such as F-Actin networks are known to have a storage modulus that increases with time, where the crossover from viscous to elastic behavior takes place at ∼1–10 s (68). Thus, the relevant timescale of cargo motion, ∼1 s, may be around that crossover; we expect viscoelastic effects to be present but not very large. Clearly, more accurate experimental data for the viscoelasticity of the axoplasm is required. We further note that in the noninertial (zero Reynolds number) limit considered here for given cargo velocities, the results for the displacements (rather than stresses) of the dragged molecules are independent of the viscosity. Qualitatively, in a viscoelastic medium, they should be similarly insensitive to the frequency-dependent moduli.

Another issue, addressed in detail in the Results section, is the effect of boundary conditions along the axonal membrane, namely, whether or not there is significant slip at the boundary. We find that slip may be either unimportant or important for axonal transport, depending on whether the axon should be described as effectively closed or open-ended, respectively (compare Figs. 3 a and 8). This issue, in turn, depends on the amount of water exchange across the axonal membrane. Thus, the issues of boundary slip and membrane permeability are intertwined and should be clarified in future experiments to obtain a more quantitative description of the transport studied here.

An important implication of this study relates to the interpretation of nonlocal measurements of intracellular diffusion of various proteins, such as by FRAP or diffusion-weighted NMR (69). It should be taken into account that intracellular diffusion coefficients extracted from such measurements in vital cells may be overestimated due to the contribution of active mechanisms that may induce microstreaming. Experiments that involve fixation of cells or blockage of various possible active processes (35) can provide the needed discrimination between diffusion and microstreaming. Only with additional experiments of this sort will we be able to quantify the relative contribution of thermal diffusion and active mechanisms in defining the SC and intracellular displacement of soluble proteins.

The implications of water displacement in neurons extend beyond the basic understanding of transport and axonal physiology. Diffusion tensor imaging (DTI) uses water displacement in the brain as a unique tool for diagnosis and research (70). Water displacement in neuronal projections is anisotropic (with preferred displacement in the longitudinal direction), and significantly drops during events of metabolic or oxidative stress, as occurs, for example, in cerebral artery occlusion (71). DTI measures water displacement over typical timescales of 1–100 ms. Our calculation implies that, contrary to a previous suggestion (72), within these timescales, the displacement of water molecules, caused by drag due to transported cargoes, is too minor to be detected by DTI. On a more general level, it should be noted that some of the displacement of water molecules, as well as the drop in displacement detected after tissue insults, may be due to blockage of intracellular microstreaming. The existence of such active microstreaming, as well as its biophysical basis, should be further investigated.

Transport mechanisms similar to those proposed here might be relevant to the mixing and distribution of solutes in other scenarios. A particular example is that of cytoplasmic streaming in plant cells (56,73–75). Organelle-induced flow is observed in many plant cells in which different flow patterns with variable velocity distributions have been identified. Nonetheless, organelle-induced flow is expected to be most prominent in cells of anisotropic cytoskeleton, where there are displacements over long distances. These conditions are met in neurons.