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Nat Cell Biol. Author manuscript; available in PMC 2010 May 11.

Published in final edited form as:

Published online 2009 September 20. doi: 10.1038/ncb1965

PMCID: PMC2867054

NIHMSID: NIHMS197439

Kinneret Keren,^{1,}^{2,}^{7} Patricia T. Yam,^{1,}^{3} Anika Kinkhabwala,^{4} Alex Mogilner,^{5} and Julie A. Theriot^{1,}^{6}

AUTHOR CONTRIBUTIONS

K.K. and J.A.T. conceived and designed the experiments; K.K. performed the experiments and analysed the data; A.M. and K.K. developed the model; P.T.Y. made the initial observation of enhancement of large probes at the leading edge; A.K. contributed to the single-particle tracking experiments; K.K., A.M., P.T.Y. and J.A.T. discussed the results and wrote the paper.

The publisher's final edited version of this article is available at Nat Cell Biol

See other articles in PMC that cite the published article.

Cytosolic fluid dynamics have been implicated in cell motility^{1}^{–}^{5} because of the hydrodynamic forces they induce and because of their influence on transport of components of the actin machinery to the leading edge. To investigate the existence and the direction of fluid flow in rapidly moving cells, we introduced inert quantum dots into the lamellipodia of fish epithelial keratocytes and analysed their distribution and motion. Our results indicate that fluid flow is directed from the cell body towards the leading edge in the cell frame of reference, at about 40% of cell speed. We propose that this forward-directed flow is driven by increased hydrostatic pressure generated at the rear of the cell by myosin contraction, and show that inhibition of myosin II activity by blebbistatin reverses the direction of fluid flow and leads to a decrease in keratocyte speed. We present a physical model for fluid pressure and flow in moving cells that quantitatively accounts for our experimental data.

Various indirect results^{4}^{,}^{6} suggest that fluid influx at the leading edge could play an active part in actin-based cell motility by generating hydrodynamic forces that oppose the membrane load and thus increase the rate of actin polymerization at a protruding edge. Notably, the expression of aquaporins, which are highly enriched at the leading edge and increase water permeability of the membrane, accelerates motility and increases the metastatic potential of melanoma cells^{4}^{,}^{7}. Alternatively, it has been suggested that intracellular fluid flow towards the leading edge might contribute to motility by expediting transport of actin and other soluble proteins to the leading edge^{8}. Direct measurements of fluid flow in the lamellipodia of moving cells are lacking, as methods for measuring intracellular flow^{9} typically use large organelles or particles (hundreds of nm to a few μm in diameter), which cannot penetrate the dense lamellipodial actin meshwork characterized by a pore size of about 50 nm^{10}^{,}^{11}. As shown below, smaller particles (<50 nm) that can penetrate the dense actin meshwork diffuse rapidly, severely limiting the sensitivity of fluid flow measurements based on single-particle tracking in live cells. To overcome this difficulty, we introduced a new approach in which the direction and magnitude of fluid flow are determined from measurements of the steady-state size-dependent spatial distribution of inert probes.

Fish keratocytes are a simple and widely used model system for studying the dynamics of the motility process. They are among the fastest moving animal cells, with average speeds of about 0.3 μm s^{–1}, yet their motion is extremely persistent, with hardly any change in cell shape, speed or direction over many minutes^{12}^{,}^{13}. In these cells, any intracellular fluid flow associated with motility should be persistent because of the steady-state nature of their motion. Furthermore, the broad, flat and extremely thin (~100–200 nm)^{10} lamellipodia of these cells alone are sufficient for persistent motility^{12}. We set out, therefore, to measure the fluid dynamics within the lamellipodia of moving keratocytes. We introduced fluorescent tracers into the cytosol as probes for the fluid dynamics. For these tracers to faithfully report the motion of the fluid phase of the cytoplasm, they must have minimal nonspecific interactions with the surrounding cellular milieu, in particular the actin meshwork in the lamellipodium. Polyethyleneglycol coating has been shown to be optimal in reducing nonspecific interactions in both *in vivo*^{14} and *in vitro* actin networks^{15}, so we chose methoxypolyethyleneglycol-coated quantum dots (QDs) as probes.

We followed the motion of individual 655QDs in live keratocytes (Fig. 1) and observed rapid movement in the lamellipodia, characterized by an average apparent diffusion coefficient of *D* = 1.15 ± 0.5 μm^{2} s^{–1} (mean ± s.d., *D* = MSD(t)/4t with MSD(t) = δr(t)^{2} where δr(t) is the QD displacement during the time interval t = 0.3 s; *n* = 37 cells). We found considerable cell-to-cell variability in the rate of apparent QD diffusion (Fig. 1c–e), probably reflecting differences in the density and organization of the lamellipodial actin meshwork among cells^{16}^{,}^{17}. Fluid flow should induce a bias in the motion of tracer particles as measured by single-particle tracking. In principle, even a small persistent flow could be detected by averaging over a large number of particles and/or over extended periods of time. However, in practice, several limitations set a lower limit on the magnitude of fluid flow that we can detect in a moving keratocyte by single-particle tracking^{18} (Supplementary Information, Supplementary Text B2). Within this detection limit (~0.2 μm s^{–1}) we did not observe a persistent bias in QD motion in the cell frame of reference in keratocytes (*n* = 10 cells; Supplementary Information, Fig. S1).

Single-particle tracking of QDs in the lamellipodia of live keratocytes. (**a**) Phase-contrast (top) and fluorescence (middle; white line: cell outline) images of a keratocyte with 655QDs. The QD trajectories during 100 frames (*dt* = 0.15 s) are shown in **...**

To increase the sensitivity of our fluid flow measurements in the lamellipodium, we turned to measuring flow-induced effects on the size-dependent distribution of inert probes. Consider a simplified case in which the lamellipodium is approximated as a thin rectangular box, and persistent fluid flow occurs along its posterior–anterior axis. We assume that the membrane water permeability is concentrated at the leading edge, as has been shown in other cell types^{4}^{,}^{5}. In this case, the fluid flow *V _{f}* will be uniform, and the probe distribution

Flow-induced size-dependent distribution of probes. (**a**) The expected distribution of a large probe diffusing with *D* = 1 μm^{2} s^{–1} as a function of distance from the leading edge for different flow rates along the anterior–posterior **...**

To measure fluid flow in moving keratocytes, we introduced large inert probes into the cytoplasm of live cells and determined their distribution in the lamellipodium, using ratio imaging with a small probe that acted as a volume marker to correct for thickness variations. In addition to the QDs, we also used large fluorescent dextrans, which have been previously used to characterize size-dependent intracellular motion in live cells^{19}, and have little nonspecific interaction. We tested several large probes (diameter >20 nm; 655QDs (30.5 ± 1 nm), 565QDs (23 ± 2 nm), 500 kD dextran) in various combinations with several small probes (diameter <1 nm; AlexaFluor488 free dye, AlexaFluor594 free dye, 3K dextran). Irrespective of the combination of probes used, we found enhancement of large probes relative to small probes towards the leading edge (Fig. 2c, d; Supplementary Information, Figs S2, S3), consistent with a forward-directed flow in the cell frame of reference, from the cell body towards the leading edge.

To estimate the magnitude of fluid flow, we measured the distribution of the 655QDs density, for which we had already measured the apparent diffusion coefficient (Fig. 1). The ratio density along a cross-section perpendicular to the leading edge (Fig. 2c, bottom panel) was fitted to an exponential profile *n(x)* = *n*_{0} exp(–*x*/*L*) (Supplementary Information, Fig. S4). The average magnitude of the fluid flow can be estimated from the average length scale 1/*L* = 0.10 ± 0.02 μm^{–1} (mean ± s.e.m., *n* = 30 cells) obtained from this fit, together with the average apparent diffusion coefficient, *D* = 1.15 ± 0.1 μm^{2} s^{–1} (mean ± s.e.m. at *t* = 0.3 s, *n* = 37 cells), giving *V _{f} ~D/L~*0.11 μm s

To understand the origin of this flow, we developed a physical model describing the behaviour of the cytosolic fluid in a moving lamellipodium (Fig. 3a). In the moving cell frame of reference, let ${\overrightarrow{V}}_{a}$ be the actin network retrograde flow rate. The relative velocity between the actin meshwork and the fluid phase is proportional to the pressure gradient, according to the Darcy flow equations1,3: ${\overrightarrow{V}}_{f}-{\overrightarrow{V}}_{a}=-K\nabla P$, where ${\overrightarrow{V}}_{f}$ is the fluid velocity, *P* is the hydrostatic pressure, and *K* is the cytoskeletal permeability. As the actin meshwork in keratocytes is nearly stationary relative to the substrate^{20}^{,}^{21}, we assumed that ${\overrightarrow{V}}_{a}$ is constant; its magnitude is equal to cell speed and it is directed rearward. The perpendicular centripetal actin network flow at the cell rear^{20}^{–}^{22} did not have a significant effect on fluid flow in the lamellipodium (Supplementary Information, Fig. S5) and was therefore disregarded.

Measurements of fluid flow based on the distribution of 655QDs. (**a**) An illustration of a keratocyte in the cell frame of reference depicting the model parameters and variables. Side (left) and top (right) views are shown. (**b**, **c**) The density distribution **...**

The permeability of the lamellipodial membrane is concentrated at the leading edge^{4}^{,}^{5}, so fluid transport across the membrane will be concentrated there. At the front boundary, the fluid outflux is proportional to the pressure drop across the membrane: ${\overrightarrow{V}}_{f}\cdot \overrightarrow{n}{\mid}_{front}={k}_{m}(P-{P}_{out}){\mid}_{front}$, where $\overrightarrow{n}$ is a unit vector locally normal to the front boundary, and *k _{m}* is the membrane permeability

$${\overrightarrow{V}}_{f}-{\overrightarrow{V}}_{a}=-K\nabla P;\nabla \cdot {\overrightarrow{V}}_{f}=0;{\overrightarrow{V}}_{f}\cdot \overrightarrow{n}{\mid}_{front}={k}_{m}(P-{P}_{out}){\mid}_{front};P{\mid}_{rear}={P}_{r}$$

(1)

Solving this model analytically in 1D (Supplementary Information, Supplementary Text A1) gives an expression for the fluid flow speed *V _{f}* as a function of the parameters

$${V}_{f}=\frac{{k}_{c}{k}_{m}}{{k}_{c}+{k}_{m}}({P}_{r}-{P}_{out})-\frac{{k}_{m}}{{k}_{c}+{k}_{m}}{V}_{a}$$

(2)

where *k _{c}* =

The full 2D model was solved numerically (Fig. 3c–e; Supplementary Information, Supplementary Text A2). The hydrostatic pressure drops from rear to front and sides (Fig. 3d) and drives a fountain-like fluid flow against the drag of the actin cytoskeleton (Fig. 3e). Using the measured diffusion rates (Fig. 1), the model predicts the size-dependent probe distribution in 2D (Fig. 3c), which can be compared directly to our experimental results (Fig. 3b). The model-based distribution nicely captured the observed enhancement of the large probe near the leading edge, as well as its accumulation at the lamellipodial wings.

We hypothesized that the increased pressure at the rear of the lamellipodium results from myosin activity. Myosin II is known to localize towards the rear of the lamellipodium in keratocytes^{11}^{,}^{24} and contract the actin meshwork. This contraction creates a compressive stress on the actomyosin network, which can lead to an increase in the local hydrostatic pressure^{1}. Inhibition of myosin II would therefore be expected to lead to a reduction in the hydrostatic pressure at the rear of the lamellipodium and, thus, according to the model (equation (2), with *P _{r}* –

Effects of blebbistatin on fluid flow. (**a**) Confocal ratio imaging of a cell treated with blebbistatin. The distribution of small probe (upper image), large probe (centre image) and the ratio between large probe to small probe (lower image) are shown. **...**

The full 2D model was solved for a blebbistatin-treated cell by simply changing the boundary condition for the pressure along the rear to *P _{r}* =

Staining of filamentous actin showed that blebbistatin did not alter the morphology of the actin network near the leading edge (Fig. 4e). Moreover, single-particle tracking experiments in live blebbistatin-treated cells did not reveal any substantial differences with respect to QD655 behaviour (Fig. 1e). Furthermore, recent work^{21} has shown that blebbistatin treatment does not change the characteristics of the actin network flow in the front lamellipodium. These observations further support the notion that the differences in QD distribution seen in blebbistatin-treated cells arise from changes in fluid dynamics, rather than alteration of the actin meshwork.

The contribution of myosin-dependent forward-directed fluid flow to the motility process has been postulated for a long time^{2}^{,}^{8}^{,}^{26}. In motile keratocytes, we found a forward-driven fluid flow caused by myosin activity, with a magnitude of about 40% of cell speed (~0.11 μm s^{–1}) in the cell frame of reference. Theoretical estimates (Supplementary Information, Supplementary Text A5, A10) suggest that this forward-directed fluid flow expedites actin monomer transport to the leading edge, and slightly increases the hydrostatic pressure there, effects which could account for a significant (~20%) increase in cell speed in untreated cells, compared with myosin-inhibited cells. These estimates are consistent with our observations that movement of blebbistatin-treated cells is about 40% slower than untreated cells; we hypothesize that about half of this effect is directly related to the myosin-driven forward-directed fluid flow, whereas the other half is related to the role of myosin in reducing membrane tension and in increasing the actin monomer concentrations by promoting actin disassembly at the rear^{20}^{,}^{21}. Taken together, these results indicate that forward-directed fluid flow is not essential for cell motility, yet flow can accelerate the motility process. Finally, any forward (rearward) directed fluid flow in the lamellipodium generates an outflux (influx) at the leading edge that must be balanced by a corresponding influx (outflux) at the cell rear. As discussed in the Supplementary Information, actin–myosin contraction can draw fluid from the surroundings into the cell body, from where fluid can then flow into the lamellipodium. As the surface area at the rear of the cell is an order of magnitude larger than that at the leading edge, the associated fluid flow at the rear would be undetectable.

The existence of a myosin-generated pressure gradient has been documented in other motile cells^{26}. Furthermore, observations in fibroblasts point to the existence of myosin-dependent rapid transport of actin towards protruding regions at the leading edge^{8}, which was hypothesized to be at least partially driven by fluid flow towards the edge. The direction of rapid actin transport observed^{8} is consistent with our results in keratocytes, whereas the magnitude of active transport inferred (up to 5 μm s^{–1}) is substantially higher than our results in keratocytes (~0.1 μm s^{–1}). This difference might be caused by actin-specific myosin-dependent active transport mechanisms, in addition to fluid flow, or alternatively may reflect differences in the dynamics of the motility process in these substantially slower-moving cells. The relative contribution of fluid flow to motility will depend on the particular dynamics of the system and on the balance between pressure-mediated and assembly-mediated protrusion. These characteristics will vary among cell types, and even among individual cells within a single cell type^{17} or individual cells in a changing environment (Supplementary Information, Fig. S6). In cases where membrane permeability at the leading edge is low, theory predicts that the pressure at the leading edge would increase and this could significantly assist other force generators there. Alternatively, high membrane permeability at the leading edge, as in keratocytes, relieves this pressure but leads to an increased forward-directed fluid flow which accelerates transport to the leading edge. Importantly our work suggests that in both cases the effect of myosin-generated contractile pressure at the rear is beneficial for motility^{4}^{,}^{7}.

Although improving technologies have enabled detailed observation of live keratocytes undergoing rapid actin-based motility, and measurement of traction^{27}^{,}^{28} and protrusive forces^{29} as well as the dynamic behaviour of specific elements of the motile machinery including actin^{20}^{,}^{22}, myosin^{11}, components of the membrane^{13} and adhesive contacts^{30}, the fluid remained an invisible part of the puzzle. There have been suggestions in the literature for decades^{2} asserting the importance of the fluid in cell motility, but experimental observations have been lacking. Here we show for the first time that fluid dynamics in moving cells can actually be measured, and this opens the way towards understanding the role of fluid flow in actin-based cell motility.□

Methods and any associated references are available in the online version of the paper at http://www.nature.com/naturecellbiology/

**Figure S1** Analysis of bias in QD tracks in a moving keratocyte. A histogram of QD displacements parallel (red, <Δx>/dt=0.22±0.15μm/s, mean±S.E.M.) and perpendicular (blue, <Δy>/dt=-0.05±0.15μm/s) to the direction of motion in the lab frame of reference, from a single cell moving with V_{cell}=0.12±0.01μm/s. Displacements are shown for time interval=38ms (upper panel), 75ms (middle panel) and 150ms (lower panel). The green line depicts the expected Gaussian distribution of displacements for purely diffusive motion with D=0.83μm^{2}/s.

**Figure S2** Size-dependent probe distributions in moving cells. The normalized ratio density profiles (averaged over a width of 2μm) along a cross section in the central lamellipodium are plotted as a function of distance from the leading edge. Profiles taken from images of different cells are superimposed. Top: the ratio of large probe to small probe (655QDs/ AF488) in untreated cells exhibits enhancement of large probe near the leading edge (N=30 cells). Center: the ratio of large probe to small probe (655QDs/AF488) in blebbistatin-treated cells typically shows depletion of large probe near the leading edge (N=25 cells). Bottom: control experiments with two small probes (AF594/AF488) exhibit a flat profile as expected (N=10 cells). The bold lines and shaded regions represent the mean and standard deviation (respectively) of the profiles of all cells in each graph.

**Figure S3** Probe distribution as a function of time. Images depicting the ratio of large probe to small probe (655QDs/ AF488) at two different time points for a single moving cell. The image on the right was taken 30 s after the image on the left (the outline of the cell on the left is overlaid on the image on the right). The cell exhibits a nearly steady-state distribution of probes characterized by enhancement of the large probe near the leading edge and in the wings.

Figure S4. Characterization of the distribution of 655QDs in moving cells. (**a**) The profile of the ratio of large probe to small probe (655QDs/AF488) as a function of distance from the leading edge along a cross section in the central lamellipodium of a moving cell (Fig.2d, bottom panel), was fitted to an exponential distribution *n(x) = n*_{0} exp(–*x*/*L*) in the region indicated, with the length scale *L* as the fit parameter. Note that *L* does not depend on the total intensity of the signal, but only on the shape of the observed ratio profile. Solid line: ratio data; Dashed line: fit. (**b**) A histogram of the values of *1*/*L* obtained in this manner from the ratio distribution (655QDs/ AF488) in a population of N=30 cells. The variation among cells can be attributed to differences in cell speed, in the pressure at the rear of the cell, and in the membrane and cytoskeletal permeabilities (further supported by the observation of variable apparent diffusion as shown in Fig. 1c-e). (**c**) The average inverse length scale *1*/*L* (mean±S.E.M.) characterizing the ratio distribution (655QDs/ AF488) in untreated and blebbistatin-treated cells. Untreated cells are characterized by enhancement of large probe at the leading edge (i.e. 1/L > 0); while blebbistatin-treated cells are characterized by depletion of large probe near the leading edge (i.e. 1/L < 0).

**Figure S5** The effect of centripetal actin network flow on fluid flow. The 2D model was solved numerically as described in the text in the presence or absence of centripetal actin network flow at the rear. The results shown in (a) were obtained under the simplifying assumption that the actin network retrograde flow is rearward and constant (top left panel), whereas the results shown in (b) take into account a more realistic actin network flow pattern with inward flow at the rear (top right panel). Using these actin network flow patterns and identical boundary conditions, the pressure field, fluid flow field, and probe density distributions (bottom panels) were calculated numerically in both cases (see Supplementary Text A3 for details). The centripetal actin flow causes an increase in the pressure in the lamellipodium so the front-to-back pressure gradient decreases, but the effect is relatively small. The sideways component of the fluid flow decreases when the centripetal actin flow is included, because the actin flow at the rear drags the fluid inward and partially cancels the effect of the hydrostatic pressure. This, together with the incompressibility of the fluid, implies that the forward fluid flow at the front of the lamellipodium increases ~ 20%. Hence, the changes in the calculated fluid flow pattern following the inclusion of the centripetal actin flow are predicted to cause more pronounced accumulation of large probe near the leading edge and less accumulation at the sides. Note that neither the qualitative fountain character of the predicted fluid flow pattern nor the order of magnitude of the flow rate change when the centripetal actin flow is added to the model.

**Figure S6** Local changes in osmolarity near the leading edge affect protrusion rate. A glass micropipette controlled by a micromanipulator was used to flow water or 0.9 M sorbitol near the leading edge of moving cells in culture media. A fluorescent marker (AF488, Molecular Probes) was added to allow visualization of the flow from the micropipette. An overall flow in the live-cell chamber was added to make the osmolarity gradient steeper. This flow was generated by a peristaltic pump connected to two needles spaced ~1 cm from each other ~1 mm above the coverslip. Cells were imaged with phase contrast and fluorescence (to monitor the flow from the micropipette) at 1 frame/s. Phase contrast images of cells during local perfusion of water (a) and 0.9 M sorbitol (b) near the leading edge (the shadow of the micropipette used is apparent in the images) are shown. (c,d) Kymographs showing the protrusion of the leading edge as a function of time along the black lines in a and b, respectively. The brackets on the right indicate the time during which the micropipettes were positioned near the leading edge. The dashed diagonal lines correspond to protrusion at a constant rate, characteristic of the steady state keratocyte motion. The protrusion rate increased (white arrows) when the osmolarity was decreased (c). Conversely, the protrusion rate decreased (white arrows) when the osmolarity was increased (d). Note that the protrusion rates shifted back toward their unperturbed rates after the micropipette was retracted.

**Figure S7** Effects of the “dynamic error” on the MSD. The experimental MSD (blue) is plotted as a function of time lag for data pooled from 16 cells acquired with an exposure of dt=0.015 s. The subdiffusive exponent γ=0.89±0.02 was calculated from data points *t* = *n* * *dt* 20 ≤ *n* ≤ 40. The black line depicts pure subdiffusive motion with γ=0.89, MSD~t^{γ}. The red line depicts the expected MSD for such motion (γ=0.89, true MSD~t^{γ}) when the contribution from the “dynamic error” is included. The seemingly diffusive behavior observed in the experimental data at short time scales appears to be an artifact associated with the “dynamic error”.

Supplementary Text

Intracellular fluid flow in rapidly moving cells

Kinneret Keren, Patricia T. Yam, Anika Kinkhabwala, Alex Mogilner and Julie A. Theriot

A. Supplementary model

A1. Pressure gradient and cytoplasmic flow across the lamellipodium: 1D model

In the framework of a steadily crawling cell, the actin meshwork moves rearward at a constant rate *V _{a}*, nearly equal to the cell's speed

$${V}_{f}={k}_{m}({P}_{front}-{P}_{out})$$

(1)

In the lamellipodium, the Darcy flow equation reads:

$${V}_{a}+{V}_{f}={k}_{c}({P}_{r}-{P}_{front})$$

(2)

The cytosol drifts towards the leading edge with a rate (*V _{a}* +

Solving equations (1-2), we find the leading edge pressure and fluid drift:

$${P}_{front}=\frac{{k}_{m}}{{k}_{c}+{k}_{m}}{P}_{out}+\frac{{k}_{c}}{{k}_{c}+{k}_{m}}{P}_{r}-\frac{{V}_{a}}{{k}_{c}+{k}_{m}}$$

(3)

$${V}_{f}=\frac{{k}_{c}{k}_{m}}{{k}_{c}+{k}_{m}}({P}_{r}-{P}_{out})-\frac{{k}_{m}}{{k}_{c}+{k}_{m}}{V}_{a}$$

(4)

In untreated cells, where *P _{r}* >

When myosin contraction is inhibited, it is reasonable to assume that the pressure at the rear drops to its level outside the cells, so *P _{r}* ≈

$${V}_{f}\approx -\frac{{k}_{m}}{{k}_{c}+{k}_{m}}{V}_{a}.$$

(5)

Thus the fluid in this case is predicted to flow against the direction of locomotion: the lamellipodium is like an earth-worm eating its way through the aqueous environment. Our measurements in blebbistatin treated cells (Fig. 4; Figs.S2, S4) demonstrate that in this case both cell speed and the rate of fluid flow are similar, ~ 0.15 *μm* / sec, which is an indication, according to (5), that the membrane permeability is higher than the permeability of the cytoskeleton: *k _{m}* >

We can analytically solve the stationary diffusion-drift equation for the probe's concentration along the posterior-anterior axis of the lamellipodium (in the 1D model, the concentration does not vary laterally): $D\frac{{d}^{2}n}{d{x}^{2}}+{V}_{f}\frac{dn}{dx}=0$. As the cell membrane is impermeable to the probe, the flux of the probe is equal to zero: $\left(-D\frac{dn}{dx}-{V}_{f}n\right)=0$. The solution of this simple linear differential equation gives an exponential concentration profile: *n*(*x*) exp(−*x*/*L*) where *x* is the distance from the leading edge and *L*=*D*/*V _{f}*. Thus, fluid flow toward the leading edge in the cell frame of reference causes an exponential accumulation of the probe near the leading edge, whereas flow toward the cell body (as in blebbistatin-treated cells) leads to exponential accumulation of the probe toward the rear.

A2. Pressure gradient and cytoplasmic flow across the lamellipodium: 2D model

The Darcy flow equations for the cytoplasmic fluid flow in 2D are:

$${\overrightarrow{V}}_{f}-{\overrightarrow{V}}_{a}=-K\nabla P,\nabla \cdot {\overrightarrow{V}}_{f}=0.$$

(6)

Here *P* is the pressure field, ${\overrightarrow{V}}_{f}$ is fluid velocity, and ${\overrightarrow{V}}_{a}$ is the velocity of the actin meshwork in the cell frame of reference. We assume that the actin meshwork velocity in the cell frame of reference, ${\overrightarrow{V}}_{a}$, is constant; its magnitude equal to the cell speed *V _{cell}*, and it is directed to the rear. This simplification is justified since the lamellipodial actin meshwork is known to be nearly stationary relative to the substrate

The boundary conditions for the Darcy flow equations are as follows. At the rear, *P* = *P _{r}* =

$${\overrightarrow{V}}_{f}\cdot \overrightarrow{n}{\mid}_{front}=({\overrightarrow{V}}_{a}-K\nabla P)\cdot \overrightarrow{n}{\mid}_{front}={k}_{m}P{\mid}_{front}\to \nabla P\cdot \overrightarrow{n}{\mid}_{front}=\frac{1}{K}({\overrightarrow{V}}_{a}\cdot \overrightarrow{n}-{k}_{m}P){\mid}_{front}$$

Thus, we first solve the pressure problem numerically:

$$\Delta P=0,P{\mid}_{rear}={P}_{r},\nabla P\cdot \overrightarrow{n}{\mid}_{front}=\frac{1}{K}({\overrightarrow{V}}_{a}\cdot \overrightarrow{n}-{k}_{m}P){\mid}_{front}$$

(7)

Then, we compute the fluid velocity as:

$${\overrightarrow{V}}_{f}={\overrightarrow{V}}_{a}-K\nabla P$$

(8)

Finally, using this velocity we numerically solve the stationary diffusion-drift equation for the probe's concentration, *C*:

$$\nabla \cdot (D\nabla C-{\overrightarrow{V}}_{f}C)=0$$

(9)

with no flux boundary conditions (i.e. the probe cannot move across the cell membrane). Note, that in 1D the no flux boundary conditions imply that there is no flux of probe inside the lamellipodium, so equation, $\nabla \cdot \left(D\nabla C-{\overrightarrow{V}}_{f}C\right)=0$, can be integrated and yields the simpler equation $D\nabla C-{\overrightarrow{V}}_{f}C=0$ which describes the probe's density. However, in 2D, the picture is not so simple: there can be circular fluxes, and in fact, we find that there are such small circular fluxes near the lamellipodial sides.

We solved equations (7-9) numerically using the Virtual Cell (http://vcell.org) biological modeling framework^{4}. The corresponding 2D model is publicly available in the Virtual Cell. The lamellipodial geometry was defined by the front and rear boundaries which were approximated with elliptical arcs; the aspect ratio of the virtual lamellipodium was close to 3. We used the constant pressure at the rear, *P _{r}*, as the unit of pressure, the lamellipodial width,

The simulation of an unperturbed cell (Fig. 3) shows a pressure decrease from rear to front (Fig. 3d) creating a forward-directed fluid flow moving against the retrograde actin meshwork flow (in the moving cell frame of reference; Fig. 3e). The fluid flow is not exactly forward; rather it is centrifugal from the cell rear to the front boundary(Fig. 3e), so that near the sides there is a large sideways component of the flow of fluid seeping through the lamellipodial sides outward. Because of this centrifugal flow pattern, the large probe density increases not only from rear to front but also from center to sides (Fig. 3c). Note the good qualitative fit between the experimental and theoretical probe density distributions. Both measured and predicted (Fig. 3b-c) posterior-anterior densities have linear profiles. This implies that the forward component of the fluid velocity at the center actually decreases from rear to front, since a constant rearward fluid velocity (as predicted by the 1D model) would generate an exponential, rather than linear, probe density profile. Note also that the 2D theory captures the observed significant increase of the large probe density toward the lamellipodial sides (Fig. 3b-c).

To simulate a blebbistatin-treated cell, we simply changed the boundary condition for the pressure along the rear to *P _{r}* =

A3. 2D results in the presence of centripetal actin network flow

To investigate the effect of centripetal actin network flow on the fluid flow in the lamellipodium^{1}^{, }^{2}, we added the centripetal flow to the actin retrograde flow as follows (Fig. S5):

$${\overrightarrow{V}}_{a}=-{V}_{a}\times \overrightarrow{j}-{V}_{a}\times \frac{x}{W}\times \text{exp}(-y\u2215l)\times \overrightarrow{i}+{V}_{a}\times \text{exp}(-y\u2215l)\times \overrightarrow{j}.$$

The first term describes a constant rearward flow; its speed is *V _{a}* = 0.25

To compute the resulting fluid flow, we differentiate equation (6): $\nabla \cdot \left({\overrightarrow{V}}_{f}-{\overrightarrow{V}}_{a}\right)=-K\nabla P$ and use the incompressibility condition $\nabla \cdot {\overrightarrow{V}}_{f}=0$ to obtain $K\Delta P=\nabla \cdot {\overrightarrow{V}}_{a}$. The divergence of the first, constant term in the expression for ${\overrightarrow{V}}_{a}$ is equal to zero, but the divergence of the second and third centripetal terms is equal to the function $-{V}_{a}\left(\frac{1}{l}+\frac{1}{W}\right)\times \text{exp}(-y\u2215l)$. Thus, instead of the equation Δ*P* = 0 for the pressure, we solve the following equation:

$$\Delta P+\frac{{V}_{a}(l+W)}{KlW}\times \text{exp}(-y\u2215l)=0.$$

The boundary conditions in the presence of centripetal actin flow are unchanged. We solved this equation numerically as described in the previous section. Then, we compute the fluid velocity using equation (8) and the probe's concentration using equation (9). These calculations lead to the results shown in Fig. S5b. Comparison of the pressure distributions with and without the centripetal actin flow, shows that the additional actin flow increases the pressure in the lamellipodium making it closer to the pressure in the cell body and decreasing the pressure gradients. The reason is clear: the centripetal actin flow imposes inward drag on the fluid that increases the pressure. However, the effect is relatively small; differences in the pressures with and without centripetal actin flow are maximal at the center of the leading edge, where they amount to ~ 30%. Elsewhere, they are much smaller.

Fig. S5 also shows the predicted velocities of the fluid flow. The additional centripetal actin flow leads to a decrease in the sideways component, *V _{x}*, of the fluid flow, with the largest change (~ 30%) at the rear sides of the lamellipodium because the centripetal flow drags the fluid inward partially canceling the effect of the hydrostatic pressure. Similarly, the forward/backward component,

The bottom panels in Fig. S5 show the predicted probes’ distribution. As intuitively expected the concentration decreases at the lamellipodial sides and increases at the middle of the lamellipodial front, because the centripetal actin flow drags the fluid and probe with it from the sides into the center. The differences in concentrations are at most ~ 30%. Note that neither the qualitative fountain character of the predicted fluid flow nor the order of magnitude of the flow rate change.

A4. Estimates of the membrane and hydraulic permeabilities

The hydraulic permeability can be estimated as ${k}_{c}=\frac{{k}_{c}^{\prime}}{\varphi {L}_{0}}$ using the formula for the hydraulic conductivity of the cytoskeleton ${k}_{c}^{\prime}\approx \frac{{l}^{2}}{\eta {\varphi}^{1\u22153}}{\phantom{\rule{thickmathspace}{0ex}}}^{5}$, where *l* ~ 50*nm* is the effective pore (mesh) size in the cytoskeleton, $\eta ={10}^{-2}pN\cdot \text{sec}\u2215\mu {m}^{2}$ is the effective viscosity of the cytosol, and *ϕ* ~ 0.5 is the cytoskeletal volume fraction. Thus, the order of magnitude of the hydraulic permeability is:

$${k}_{c}\approx \frac{{l}^{2}}{\eta {L}_{0}{\varphi}^{4\u22153}}~\frac{{\left(0.05\mu m\right)}^{2}}{({10}^{-2}\phantom{\rule{thickmathspace}{0ex}}pN\cdot \text{sec}\u2215\mu {m}^{2})\times 10\mu m\times {0.5}^{4\u22153}}~{10}^{-2}\frac{\mu {m}^{3}}{pN\cdot \text{sec}}.$$

(10)

The membrane permeability can be estimated using the formula ${k}_{m}=\frac{{k}_{osm}{v}_{w}}{RT}{\phantom{\rule{thickmathspace}{0ex}}}^{5}$, where R = 8.3*j* / *mol* · *K* is the universal gas constant, *T* ≈ 300*K* is the absolute temperature, *v _{w}* = 20

$${k}_{m}~\frac{(10\mu m\u2215\text{sec})\times (20ml\u2215mol)}{(8.3J\u2215mol\cdot K)\times \left(300K\right)}~{10}^{-7}\frac{\mu {m}^{3}}{pN\cdot \text{sec}}.$$

Thus, membrane permeability for pure lipid membranes is five orders of magnitude less than the cytoskeletal permeability. Aquaporins increase the membrane conductivity up to two orders of magnitude^{6} (the osmotic water conductivity becomes *k _{osm}* > 10

$${k}_{m}~\frac{({10}^{3}\mu m\u2215\text{sec})\times (20ml\u2215mol)}{(8.3J\u2215mol\cdot K)\times \left(300K\right)}~{10}^{-5}\frac{\mu {m}^{3}}{pN\cdot \text{sec}}.$$

(11)

Note that this estimate gives the average permeability over the entire cell surface. It is known that the membrane structure and biophysical properties at the leading edge of keratocytes are different from those elsewhere in the cell^{7}. Moreover, preferential localization of aquaporins at the leading edge of moving cells has been observed^{3}. Such localization would lead to a non-uniform distribution of membrane permeability to water across the cell surface, with exceptionally high permeability at the very leading edge of the motile cell. In order for the cytoplasmic flow to be of the measured order of magnitude, the cytoskeletal permeability and the membrane permeability at the leading edge have to be of the same order of magnitude, ${k}_{c}~{k}_{m}~{10}^{-2}\frac{\mu {m}^{3}}{pN\cdot \text{sec}}$, suggesting that indeed the leading edge in keratocytes is characterized by substantially higher permeability than other regions of the cell membrane. As discussed below, this is further supported by analysis, which indicates that the observed fluid flow pattern is inconsistent with a uniform distribution of membrane permeability along the dorsal surface.

A5. Effects of the hydrostatic pressure associated with fluid flow

Using the assumption *k _{m}* ≈ 3

$$\frac{{k}_{c}{k}_{m}}{{k}_{c}+{k}_{m}}({P}_{r}-{P}_{out})=0.75{k}_{c}({P}_{r}-{P}_{out})={V}_{f}+\frac{{k}_{m}}{{k}_{c}+{k}_{m}}{V}_{0}={V}_{f}+0.75{V}_{0}({P}_{r}-{P}_{out})=\frac{{V}_{f}+0.75{V}_{0}}{0.75{k}_{c}}\approx 35\frac{pN}{\mu {m}^{2}}$$

(12)

in untreated cells. This estimate fits well with various indirect estimates in the literature of the magnitude of the contractile stress at the lamellipodial rear^{8}^{, }^{9}, all of the order of $~100\frac{pN}{\mu {m}^{2}}$. From equation (1), we find that in this case the pressure at the leading edge is:

$${P}_{front}={P}_{out}+\frac{{V}_{f}}{{k}_{m}}~{P}_{out}+3\frac{pN}{\mu {m}^{2}}$$

(13)

so the flow caused by contraction at the rear augments ratchet-generated protrusive force. The respective fluid-induced contribution is small – according to both theoretical estimates, and direct measurement, the characteristic actin growth-generated protrusive pressure at the leading edge is $~{10}^{3}\frac{pN}{\mu {m}^{2}}$. However, measurements^{9} illustrate that the rate of protrusion can be very sensitive to small changes in the protrusive force. It is worth noting that in blebbistatin-treated cells, the pressure drops to:

$${P}_{front}\approx {P}_{out}-\frac{{V}_{0}}{{k}_{m}}~{P}_{out}-10\frac{pN}{\mu {m}^{2}},$$

and this drop may contribute to the observed slowing down of blebbistatin-treated cells.

Finally, note that the height of the lamellipodium does not change significantly from front to rear. Considering the elevated hydrostatic pressure of the cytoplasm, this implies that the dorsal membrane surface has to be mechanically associated with the underlying actin network connected to the ventral surface through adhesions. The lamellipodial actin network rigidity (Young modulus $~{10}^{4}\frac{pN}{\mu {m}^{2}}{\phantom{\rule{thickmathspace}{0ex}}}^{10}$) is high enough to withstand the characteristic hydrostatic pressure of tens of $\frac{pN}{\mu {m}^{2}}$.

A6. Membrane permeability is concentrated at the leading edge

The following calculation presents a strong theoretical argument in favor of the hypothesis that the membrane is much more permeable at the leading edge than at the dorsal surface. Let us denote the permeability per unit area of the dorsal surface by *k _{d}*. In 1D, let

$$\frac{dv}{dx}=\frac{{k}_{d}}{h}[{P}_{out}-p\left(x\right)],$$

(14)

where *h* is the height of the lamellipodium. The pressure gradient along the lamellipodium can be found from the Darcy flow equation:

$$\frac{dp}{dx}=\frac{-1}{{k}_{c}^{\prime}}[v\left(x\right)-{V}_{a}].$$

(15)

Differentiating (14) and substituting (15) into it, we obtain the following equation: $\frac{{d}^{2}{v}^{\prime}}{d{x}^{2}}=\frac{{k}_{d}}{{k}_{c}^{\prime}h}{v}^{\prime}$, where *v*' = *v*(*x*) − *V _{a}*. The biologically relevant solution predicts an exponential decrease of the flow rate in space from the rear toward the front of the lamellipodium:

$$v\left(x\right)~\text{exp}[x\u2215l],\phantom{\rule{1em}{0ex}}l~\sqrt{{k}_{c}^{\prime}h\u2215{k}_{d}}$$

(16)

on a spatial scale given by $l~\sqrt{{k}_{c}^{\prime}h\u2215{k}_{d}}$. This implies that unless the dorsal surface permeability *k _{d}* is small enough, the flow velocity should be noticeable only at the very rear of the lamellipodium, in which case the probe distribution across most of the lamellipodium would be uniform on average. This is inconsistent with our observations (Figs.2,,3).3). Therefore, the parameter

A7. Inhomogeneities and their effect on fluid flow and probe distribution

Let us demonstrate, first, that the probe density fluctuations depend mostly on spatial inhomogeneities of the diffusion coefficient (mostly due to structural variations in the actin meshwork), and not on fluctuations of the fluid velocity. Let us write the stationary diffusion-drift equation for the probe concentration: $\nabla \cdot \left(D\nabla C-{\overrightarrow{V}}_{f}C\right)=0$ in the form: $\nabla \cdot \left((\stackrel{\u2012}{D}+d)\nabla (C+c)-(\overrightarrow{V}+\overrightarrow{v})(C+c)\right)=0$, where *D̄* is the average constant diffusion coefficient, *d* is the small variable part of the diffusion coefficient, *C* is the computed smooth probe density distribution,*c* is the small fluctuating part of the density, $\overrightarrow{V}$ is the computed smooth fluid velocity, and $\overrightarrow{v}$ is the small fluctuating part of the velocity. Let us consider the spatial inhomogeneities on the short, micron scale, compared to the long, tens of microns lamellipodial size, and assume that the fluctuations of the diffusion coefficient, probe density and velocity are relatively small. Then, in the linear approximation, the following equation governs the density fluctuations: $\stackrel{\u2012}{D}\Delta c-\overrightarrow{V}\cdot \nabla c\approx -(\nabla C)\cdot (\nabla d)$. In other words, inhomogeneities of the velocity only perturb the density fluctuations in an insignificantly small way compared to the inhomogeneities of the diffusion coefficient, provided that the velocity fluctuations are not much greater than those of the diffusion coefficient. The estimate below suggests that indeed the velocity fluctuations are of the same order of magnitude as inhomogeneities of the diffusion coefficient.

Simply speaking, equation $\stackrel{\u2012}{D}\Delta c-\overrightarrow{V}\cdot \nabla c\approx -(\nabla C)\cdot (\nabla d)$ predicts that local maxima of the probe density should correspond to local minima of the diffusion coefficient, and visa versa. Local minima (maxima) of the diffusion coefficient should correspond to the local maxima (minima) of the actin meshwork density. This implies that the spatial fluctuations of the actin meshwork and the large probe density should correlate. The estimate below also shows that spatial fluctuations of the fluid flow are determined mainly by spatial fluctuations of the diffusion coefficient, and ultimately, of the actin density. However, there is no direct correspondence between the maxima (minima) of the flow speed and those of the actin filament density, so there is no easy way to predict the inhomogeneities of the fluid flow. Nevertheless, it is unlikely that there is a channel-like flow in the lamellipodium, because the estimates below suggest that the flow speed fluctuations are of the same order of magnitude as those of the actin filament density, and the latter are not great.

A8. Estimate of fluid flow fluctuations

Let us assume that the permeability of the cytoskeleton varies from point to point: $\overrightarrow{V}={\overrightarrow{V}}_{a}-K(x,y)\nabla P,\nabla \cdot \left(K(x,y)\nabla P-{\overrightarrow{V}}_{a}\right)=0$, and that there is a small variable component of the actin flow velocity: ${\overrightarrow{V}}_{a}={\overrightarrow{V}}_{a}^{const}+{\overrightarrow{v}}_{a}={\overrightarrow{V}}_{a}^{const}+{v}^{\left(1\right)}(x,y)\overrightarrow{i}+{v}^{\left(2\right)}(x,y)\overrightarrow{j}$, where $\overrightarrow{i}$ and $\overrightarrow{j}$ are the unit vectors in the direction of the x- and y-axis, respectively. Then, $\nabla \cdot {\overrightarrow{V}}_{a}={\stackrel{~}{v}}_{a}=\frac{\partial {v}^{\left(1\right)}}{\partial x}+\frac{\partial {v}^{\left(2\right)}}{\partial y}$. Rewriting the Darcy flow equation in the form: $\frac{\partial}{\partial x}\left(K\frac{\partial P}{\partial x}\right)+\frac{\partial}{\partial y}\left(K\frac{\partial P}{\partial y}\right)={\stackrel{~}{v}}_{a}$, and separating the small fluctuating parts of the permeability and pressure from their smooth distributions: *K* (*x*, *y*) = *K̄* + *k* (*x*, *y*), *P*(*x*, *y*) = *P*_{0} −α*y* + *p*(*x*, *y*), we can write the linear approximation for the variable part of the pressure:

$$\stackrel{\u2012}{K}\left(\frac{{\partial}^{2}p}{\partial {x}^{2}}+\frac{{\partial}^{2}p}{\partial {y}^{2}}\right)\approx \alpha \frac{\partial k}{\partial y}+{\stackrel{~}{v}}_{a}.$$

Expanding the fluctuations of the variables into Fourier series (here we do not do the analysis rigorously, but only estimate the magnitude of the fluctuations, so we do not bother with the boundary conditions and limits of the series):

$$k(x,y)=\sum {k}_{q,g}{e}^{iqx}{e}^{igy},{v}^{\left(1\right)}(x,y)=\sum {v}_{q,g}^{\left(1\right)}{e}^{iqx}{e}^{igy},{v}^{\left(2\right)}(x,y)=\sum {v}_{q,g}^{\left(2\right)}{e}^{iqx}{e}^{igy},{\stackrel{~}{v}}_{a}(x,y)=\sum (iq{v}_{q,g}^{\left(1\right)}+ig{v}_{q,g}^{\left(2\right)}){e}^{iqx}{e}^{igy},p(x,y)=\sum {p}_{q,g}{e}^{iqx}{e}^{igy}$$

we obtain the amplitude of the harmonics of the pressure fluctuations: ${p}_{q,g}\approx -i\frac{\alpha g{k}_{q,g}+q{v}_{q,g}^{\left(1\right)}+g{v}_{q,g}^{\left(2\right)}}{\stackrel{\u2012}{K}({q}^{2}+{g}^{2})}$. Substituting these into the Darcy flow equations:

$$\overrightarrow{V}={\overrightarrow{V}}_{a}-K(x,y)\nabla P={\overrightarrow{V}}_{a}^{const}+{\overrightarrow{v}}_{a}(x,y)-(\stackrel{\u2012}{K}+k(x,y))(-\alpha \overrightarrow{j}+\nabla p)\approx ({\overrightarrow{V}}_{a}^{const}+\stackrel{\u2012}{K}\alpha \overrightarrow{j})+\underset{\overrightarrow{v}}{\underbrace{({\overrightarrow{v}}_{a}(x,y)-\stackrel{\u2012}{K}\nabla p+\alpha \overrightarrow{j}k(x,y))}},$$

we find the expression for the fluctuations of the fluid velocity: $\overrightarrow{v}=\sum (\overrightarrow{i}{v}_{q,g}^{x}+\overrightarrow{j}{v}_{q,g}^{y}){e}^{iqx}{e}^{igy}$, where the amplitudes of the harmonics are given by the formulae:

$${v}_{q,g}^{x}=\frac{{g}^{2}{v}_{q,g}^{\left(1\right)}-\alpha qg{k}_{q,g}-gq{v}_{q,g}^{\left(2\right)}}{({q}^{2}+{g}^{2})},{v}_{q,g}^{y}=\frac{{q}^{2}{v}_{q,g}^{\left(2\right)}+\alpha {q}^{2}{k}_{q,g}-gq{v}_{q,g}^{\left(1\right)}}{({q}^{2}+{g}^{2})}.$$

The fluctuations of the actin network flow in space are negligible^{2} because of the very high effective viscosity of the actin gel, so respective amplitudes ${v}_{q,g}^{\left(1\right)},{v}_{q,g}^{\left(2\right)}$ can be neglected. The wave numbers *q* and *g* are of the same order of magnitude, so by order of magnitude, the fluid velocity fluctuations are simply determined by the spatial fluctuations of the permeability of the cytoskeleton: ${v}_{q,g}^{x},{v}_{q,g}^{y}~\alpha {k}_{q,g}$. The latter are likely to co-localize with the fluctuations of the diffusion coefficient. Note however, that there is no simple direct correlation between the location of the maxima (minima) of the fluid velocity and those of the permeability.

A9. Pressure and flow in the cell body

Any influx/outflux at the leading edge must be balanced by a corresponding outflux/influx at the cell rear. However, as the surface area at the rear of the cell is an order of magnitude larger than that at the leading edge (due to the much greater thickness of the cell body compared to the lamellipodium), the associated fluid flow at the rear would be undetectable. Still, any net fluid flow in the lamellipodium either toward the leading edge (under normal conditions) or away from it (when myosin is inhibited), must be balanced at the rear by influx from or outflux into the cell body, as well as the outflux/influx between the cell body and the aqueous environment around the cell. We do not analyze these processes in detail due to lack of data regarding the structure and the biophysical properties of the cell body, but discuss a few relevant issues below.

The membrane area around the cell body (roughly a half sphere with *R* ~ 5 *μm* radius) is of the order of 2*πR*^{2} ~ 150*μm*^{2}. The area of the leading edge through which the fluid flows is of the order of 25*μm*×0.15*μm* ~ 4*μm*^{2}. Therefore, in order to balance the amount of fluid flowing out of (into) the leading edge and into (out of) the cell body, the rate of flow at the cell body surface has to be ~ (4*μm*^{2} /150*μm*^{2})×0.1*μm* / sec ~ 0.0025*μm* / sec. Flow of this magnitude (just a few nanometers per second) would be undetectable. Furthermore, in order to maintain such flow, when the membrane permeability is of the order of $~{10}^{-5}\frac{\mu {m}^{3}}{pN\cdot \text{sec}}$ (see above), a pressure of the order of $~100\frac{pN}{\mu {m}^{2}}$ (characteristic for the cell rear; see above) would be sufficient. If the membrane permeability around the cell body is a few-fold higher than that, then even smaller pressures would suffice. When myosin is inhibited, the cell simply ‘moves through the fluid as an earthworm’: fluid enters the leading edge through the relatively ‘transparent’ membrane there and exits through the much larger surface area around the cell body. Under normal conditions, myosin contracts the actin cytoskeleton in the vicinity of the cell body, and this contraction pushes the cytoplasm toward a region of lower pressure at the leading edge. In addition, by contracting from the rear of the cell body toward its front ^{11}, the actin-myosin network could ‘drag’ the fluid forward in the cell body and create an effective suction at the cell body surface and thus ‘pump’ fluid into the cell body from the environment.

A10. Effect of fluid flow on transport of actin monomers and actin accessory proteins

Here we consider the actin monomer transport in a steadily moving lamellipodium of length *L*_{0}. Actin polymerization occurs mainly at the leading edge, while disassembly occurs throughout the lamellipodium ^{1}. For simplicity, let us consider the extreme case in which all actin network disassembly takes place at the very rear of the lamellipodium; in this case actin monomers have to be recycled across the whole lamellipodium, and their transport is the most sensitive to cytoplasmic flow. Then, the following reaction-diffusion equation describes the steady state actin monomer distribution in the lamellipodium^{12}:

$$D\frac{{d}^{2}g}{d{x}^{2}}+{V}_{f}\frac{dg}{dx}=0$$

(17)

where the *x*-axis is directed toward the rear and its origin is at the leading edge, *g(x)* is the actin monomer concentration, *V _{f}* is the cytoplasmic drift (positive when the flow is directed toward the front of the cell), and

$$g={A}_{1}+{A}_{2}\phantom{\rule{thinmathspace}{0ex}}\text{exp}\left[-\frac{{V}_{f}x}{D}\right]$$

(18)

The arbitrary constants *A*_{1} and *A*_{2} can be found from the following two conditions:

- The flux of actin monomers from the rear to the front, $J=D\frac{dg}{dx}+{V}_{f}g$, is constant.
- The total amount of actin monomers is regulated by the cell, $G={\int}_{0}^{{L}_{0}}g\left(x\right)dx$, and is also fixed.

Using these two conditions, we find:

$${A}_{1}=J\u2215{V}_{f},\phantom{\rule{1em}{0ex}}{A}_{2}=\frac{G+\frac{J{L}_{0}}{{V}_{f}}}{\frac{D}{{V}_{f}}\left(\text{exp}\left[\frac{{V}_{f}{L}_{0}}{D}\right]-1\right)}$$

(19)

The polymerization flux at the leading edge, *J*, is proportional to the rate of actin assembly, which in turn is proportional to the local actin monomer concentration:

$$J=kg\left(0\right)$$

(20)

where the proportionality coefficient *k* ~ 10*μm* / sec ^{12}. Combining (18-20), we estimate the polymerization flux:

$$J=kg\left(0\right)=G{\left[\frac{D}{k{V}_{f}}\left(\text{exp}\left[\frac{{V}_{f}{L}_{0}}{D}\right]-1\right)+\frac{D}{{V}_{f}^{2}}\left(\text{exp}\left[\frac{{V}_{f}{L}_{0}}{D}\right]-1\right)-\frac{{L}_{0}}{{V}_{f}}\right]}^{-1}$$

(21)

This complicated expression becomes very simple in the relevant biological limit. Indeed, we measured the diffusion coefficient of ~30nm particles in the keratocyte cytoplasm to be $~1\frac{\mu {m}^{2}}{\text{sec}}$. An actin monomer is about 4 times smaller (~ 7-8 nm), so its diffusion coefficient is at least $4\frac{\mu {m}^{2}}{\text{sec}}$, and perhaps higher, since in the crowded cytoskeleton the diffusion coefficient may grow with decreasing size slightly faster than the inverse size of the particle ^{13}. This estimate is consistent with direct measurement of the diffusion coefficient of actin monomers in other cell types, $D~5-6\frac{\mu {m}^{2}}{\text{sec}}{\phantom{\rule{thickmathspace}{0ex}}}^{14}$. Considering that *L _{0} ~ 10μm*, and ${V}_{f}<0.2\frac{\mu m}{\text{sec}}$, the ratio $\frac{{V}_{f}{L}_{0}}{D}<0.4$ is small, implying that the directed flow is weaker than diffusion as a transport mechanism. Expanding the equation for the flux in power series with respect to the factor $\frac{{V}_{f}{L}_{0}}{D}$ and keeping only the respective linear term, we can simplify the expression for the polymerization flux to:

$$J\approx \frac{2DG}{{{L}_{0}}^{2}}\left(1-\frac{2D}{k{L}_{0}}-\frac{{V}_{f}{L}_{0}}{3D}\right)$$

Here, $\frac{2D}{{{L}_{0}}^{2}}~0.1{\text{sec}}^{-1}$ gives the characteristic, diffusion-limited rate for actin monomer transport. The two small dimensionless factors in the bracket are responsible for slowing down of this transport by actin assembly and rearward cytoplasmic flow, respectively. The first factor, $\frac{2D}{k{L}_{0}}~0.1$, is small, as is the second one: $\frac{{V}_{f}{L}_{0}}{3D}<0.15$ if *V _{f}* < 0.2

It is interesting to consider the possible effect of fluid flow on the transport of other proteins necessary for the protrusion. One such important protein is Arp2/3 complex; Arp2/3 has a characteristic size of ~15nm ^{15}, and we can extrapolate from our result for QDs a diffusion coefficient of $D~2\frac{\mu {m}^{2}}{\text{sec}}$. This implies that $\frac{{V}_{f}L}{3D}<0.3$ if ${V}_{f}<0.2\mu m\u2215\text{sec}$. Larger protein complexes, ~30 nm in size, would have a diffusion coefficient of $\approx 1\frac{\mu {m}^{2}}{\text{sec}}$ comparable to the 655QDs, in which case the factor $\frac{{V}_{f}L}{3D}~\frac{2}{3}$ if *V _{f}* ~ 0.2

It is also interesting to consider the implications of the rapid fountain-like flow in the wings of the keratocyte. Our 2D simulations show that pressure varies very little at the leading edge and sides, so there is no mechanical effect of this flow. However, there is an interesting effect for the actin monomer distribution: according to our calculations, the fast flow at the sides would increase the actin monomer concentration there by ~ 50%, compared to 15-20% at the front. According to our results^{16}, the actin filaments are stalled at the sides, so the higher actin monomer concentration does not contribute to faster actin polymerization. It does contribute to the force the filaments exert on the membrane though: the stall force depends on the actin monomer concentration as ${f}_{stall}=\frac{{k}_{B}T}{\delta}\times \mathrm{ln}\left(\frac{{k}_{on}g}{{k}_{off}}\right)$, where the pre-logarithmic factor, the ratio of the thermal energy and actin monomer size, is of the order of a few piconewtons. The argument of the logarithmic function is of the order of 100 at characteristic actin monomer concentrations^{12}, so when the G-actin concentrations increases ~ 1.5-fold, the stall force increases by only ~ 10% due to the very slow growth of the logarithmic function. Therefore, the effect of fluid flow at the lamellipodial sides is weak. A small effect of this slight increase of the stall force of each filament would be that a slightly smaller filament density would be able to sustain the constant membrane tension at the sides, so according to the results of^{16}, a keratocyte has to become slightly more elongated, canoe-like. This is in agreement with results^{16}, that show that treatment of cells with blebbistatin, which according to our data abolishes the inward actin flow, makes the cell less elongated and more round. There are, actually, multiple factors causing this geometric change (discussed in ^{16}), nevertheless, the agreement between this prediction and observations is satisfying.

B. Further analysis of single particle data

B1. Analysis of the mean squared displacement as a function of time and the contribution of dynamic error

Localization error of particles in single-particle tracking experiments is known to propagate and lead to error in the mean squared displacement (MSD) as a function of time^{17}. The localization error has been separated into two contributions: a “static error” which arises in the localization of a static particle, and a “dynamic error” which comes from particle movement during the finite exposure time^{17}. For our experiments the contribution of the dynamic error is larger since the signal-to-noise from the QDs is relatively high (leading to a smaller “static error”) and their movement is relatively fast (D~1μm^{2}/s). Measurements with 15ms exposure lead to an MSD which appears diffusive (γ=0.99±0.01) on short time scales (time lag<0.1s) with a transition to subdiffusive motion (γ=0.89±0.02) on longer time scales (Fig. S7). However, closer inspection reveals that this apparent change in the characteristics of the MSD as a function of time is mostly due to the contribution of the dynamic error to the measurements, and the true motion appears subdiffusive down to at least 15ms.

Note that the associated diffusive length scale, ${l}_{D}(t=0.015s)=\sqrt{4\cdot 1\mu {m}^{2}\u2215s\cdot 0.015s}~0.25\mu m$ is substantially larger than the typical pore size of the actin meshwork in the lamellipodium ^{18}^{, }^{19}. Thus, if the observed subdiffusive behavior is related to the actin meshwork acting as obstacles for the QD motion ^{20}, motion is expected to be subdiffusive down to even smaller time scales.

B2. Analysis of bias in single-particle tracking measurements

Since the QD motion appears subdiffusive on all measured time scales it is not possible to estimate the rate of systematic transport simply from the curvature of the MSD as a function of time^{21}. However, systematic transport should still lead to a bias in the average velocity along the direction of transport. For a finite number of data points from a given cell, the detection limit for such measurements can be estimated ^{21} from the statistical error in the detected bias $\langle V\left(dt\right)\rangle =\raisebox{1ex}{$\langle \delta r\left(dt\right)\rangle $}\!\left/ \!\raisebox{-1ex}{$dt$}\right.$ which is given by $\sqrt{\raisebox{1ex}{$4D$}\!\left/ \!\raisebox{-1ex}{$N\Delta T$}\right.}$ for a diffusive process characterized by a diffusion coefficient *D*, where *N* is the total number of data points and Δ*T* is the frame acquisition time. Note that the detection limit does not strongly depend on the acquisition rate since, $N\cdot \Delta T~\#QDs\cdot \raisebox{1ex}{${T}_{total}$}\!\left/ \!\raisebox{-1ex}{$\Delta T$}\right.\Delta T=\#QDs\cdot {T}_{total}$, where *#QDs* is the number of QDs in the lamellipodium and *T _{total}* is the total acquisition time. Reasonable estimates for these parameters are

In order to estimate the bias in our single-particle data, we aligned the movies so that the cells are initially moving in the +x direction, and examined separately the displacements perpendicular and parallel to this direction in the lab frame of reference (note that while keratocyte movement is quite persistent, the cells’ velocity does deviate from their initial direction with time). Analysis of the histogram of displacements of QDs perpendicular and parallel to the direction of motion, as well as the average bias along these directions <Δy> and <Δx>, was performed on 10 moving cells. Fig. S1 depicts typical results of this analysis for one cell. The results in all cells indicate, within the limitations discussed above, the absence of uniform persistent fluid flow of magnitude larger than ~0.2μm/s in the cell frame of reference.

C. Anomalous diffusion of the quantum dots

In the main text we considered the effect of drift induced by fluid flow on the distribution of probes exhibiting diffusive motion. However, the observed QD motion is subdiffusive (Fig. 1). Here, we discuss the implications of the anomalous diffusion of the QDs on their distribution and the effect of fluid-flow induced drift on this distribution. The phenomenon of subdiffusion, where the growth of the MSD as a function of time is slower than a linear function, *r*^{2} ~ *t ^{α}, α* < 1, is ubiquitous both in cell biology and in many other biological and non-biological systems, including particles in polymeric networks

The histogram of QDs’ displacements (Fig. S1) show a very good fit with the Gaussian spread characteristic for normal diffusion, albeit with an increased number of very small displacements < 0.1 μm. This kind of histogram is very similar to that reported elsewhere^{23}, where an increased number of small displacements is the cause of slower than linear increase of MSD with time, and it could be indicative of effective transient traps in the actin meshwork. Curiously, the number of QDs exhibiting unusually small displacements decreases with time as follows: ~ 3.5% at 38 ms, ~ 2% at 75 ms and ~ 1% at 150 ms; these times can be well approximated by integrating the waiting time probability distribution $~1-{\int}^{T}\frac{dt}{{t}^{1.9}}$ which gives the fraction of the QDs that do not escape traps by *T* = 38ms, 75ms, and 150ms, respectively. (This argument by no means proves this particular scenario of subdiffusion.)

Theory indicates that with time, when the particles equilibrate with the ‘traps’ or obstacles in the cytoskeleton, the subdiffusive behavior transitions into normal diffusion (*r*^{2} ~ *t*,*t* > *t _{c}*)

Within the mathematical framework of the Continuous Time Random Walk theory, the so called Galilei-variant particle sticking model^{24} describes a situation very similar to what we assume happens with the QDs in the cytoskeleton – combined diffusion and drift intermittent with brief sticking to or trapping within the actin meshwork which is immobile in the lab coordinate system. In this situation, the probability distribution of the particles is described by the Fractional Diffusion-Advection Equation16:

$$\frac{\partial P(x,t)}{\partial t}={}_{0}D_{t}^{1-\alpha}\left(A\frac{\partial}{\partial x}+B\frac{{\partial}^{2}}{\partial {x}^{2}}\right)P(x,t),$$

where ${}_{0}D_{t}^{1-\alpha}$ is the so called Riemann-Liouville operator. This equation states that after a transient in time, the steady state distribution is governed by the usual diffusion-drift equation, $A\frac{dP\left(x\right)}{dx}+B\frac{{d}^{2}P\left(x\right)}{d{x}^{2}}=0$, that we use in our analysis of the cytoplasmic drift. It is worth noting that if the anomalous diffusion regime persisted without a crossover to the normal diffusion regime, then subdiffusion alone would be too slow to displace particles against the drift at long time scales^{24}, and one would expect all particles to concentrate at the very leading edge in a wild type cell, or at the very rear in a blebbistatin-treated cell, which is not observed.

Finally, if transient traps were responsible for the subdiffusive behavior, then the effective drift would be the actual cytoplasmic flow rate minus the rate of the actin network flow factored by the fraction of time the particles are trapped. Our observations suggest that this fraction is negligible, so the corresponding error is very small. Also, we cannot measure the diffusion coefficient in the normal diffusion regime directly, because the crossover to this regime seems to take place after 10 seconds, which is the upper limit for our MSD measurements. However, when the exponent *α* is so close to 1, as we observe, the apparent diffusion coefficient defined as^{22}:

$$\langle {r}^{2}\rangle ~4\Gamma {t}^{\alpha}=4D\left(t\right)t,D\left(t\right)=\Gamma {t}^{\alpha -1}\approx \Gamma {t}^{-0.1},$$

is a very reasonable approximation for the value of the limiting diffusion coefficient. This effective diffusion coefficient slightly decreases (it changes ± 30% between 100ms to 40s) because on longer time scales the cytoskeletal barriers restrict diffusion more and more. Therefore, we use a characteristic value of *D* ~ 1*μm*^{2} / sec for the normalized diffusion coefficient.

Supplementary References:

1. Theriot, J.A. & Mitchison, T.J. Actin microfilament dynamics in locomoting cells. *Nature ***352**, 126-131 (1991).

2. Wilson, C.A., Large scale coordination of actin meshwork flow in rapidly moving cells, *Ph.D. thesis, Stanford University*, 1-101 (2006).

3. Saadoun, S., Papadopoulos, M.C., Hara-Chikuma, M. & Verkman, A.S. Impairment of angiogenesis and cell migration by targeted aquaporin-1 gene disruption. *Nature ***434**, 786-792 (2005).

4. Slepchenko, B.M., Schaff, J.C., Macara, I. & Loew, L.M. Quantitative cell biology with the Virtual Cell. *Trends Cell Biol ***13**, 570-576 (2003).

5. Charras, G., Yarrow, J., Horton, M., Mahadevan, L. & Mitchison, T. Nonequilibration of hydrostatic pressure in blebbing cells. *Nature ***435**, 365-369 (2005).

6. Verkman, A.S. Water permeability measurement in living cells and complex tissues. *J Membr Biol ***173**, 73-87 (2000).

7. Weisswange, I., Bretschneider, T. & Anderson, K.I. The leading edge is a lipid diffusion barrier. *J Cell Sci ***118**, 4375-4380 (2005).

8. Oliver, T., Dembo, M. & Jacobson, K. Separation of propulsive and adhesive traction stresses in locomoting keratocytes. *J Cell Biol ***145**, 589-604 (1999).

9. Prass, M., Jacobson, K., Mogilner, A. & Radmacher, M. Direct measurement of the lamellipodial protrusive force in a migrating cell. *J Cell Biol ***174**, 767-772 (2006).

10. Laurent, V.M. *et al.* Gradient of rigidity in the lamellipodia of migrating cells revealed by atomic force microscopy. *Biophys J ***89**, 667-675 (2005).

11. Schaub, S., Bohnet, S., Laurent, V.M., Meister, J.J. & Verkhovsky, A.B. Comparative maps of motion and assembly of filamentous actin and myosin II in migrating cells. *Mol Biol Cell ***18**, 3723-3732 (2007).

12. Mogilner, A. & Edelstein-Keshet, L. Regulation of actin dynamics in rapidly moving cells: a quantitative analysis. *Biophys J ***83**, 1237-1258 (2002).

13. Luby-Phelps, K. Cytoarchitecture and physical properties of cytoplasm: volume, viscosity, diffusion, intracellular surface area. *Int Rev Cytol ***192**, 189-221 (2000).

14. McGrath, J.L., Tardy, Y., Dewey, C.F., Jr., Meister, J.J. & Hartwig, J.H. Simultaneous measurements of actin filament turnover, filament fraction, and monomer diffusion in endothelial cells. *Biophys J ***75**, 2070-2078 (1998).

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18. Abraham, V.C., Krishnamurthi, V., Taylor, D.L. & Lanni, F. The actin-based nanomachine at the leading edge of migrating cells. *Biophys J ***77**, 1721-1732 (1999).

19. Svitkina, T.M., Verkhovsky, A.B., McQuade, K.M. & Borisy, G.G. Analysis of the actin-myosin II system in fish epidermal keratocytes: mechanism of cell body translocation. *J Cell Biol ***139**, 397-415 (1997).

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Click here to view.^{(1.8M, pdf)}

We thank Theresa Harper and Marcel Bruchez from Quantum Dot Corporation (Molecular Probes, Invitrogen) for providing the quantum dot probes; W.E. Moerner for advice on the single-particle tracking experiments; Paul Wiseman, Ben Hebert and Lia Gracey for their contributions to the initial phase of this project; Michael Saxton and Cyrus Wilson for useful discussions; and Boris Slepchenko for help with Virtual Cell. K.K. was supported by a Damon Runyon Postdoctoral Fellowship, a Horev fellowship from the Technion, an Allon Fellowship from the Israel Council for Higher Education, and by grants from the Morasha Program of the Israel Science Foundation, the Converging Technologies Program of The Israel Council for Higher Education, the Wolfson Foundation, and a European Research Council Starting Grant. A.M. was supported by grants from the National Institutes of Health and the National Science Foundation. J.A.T. was supported by grants from the National Institutes of Health, the American Heart Association and the Howard Hughes Medical Institute. P.T.Y. was supported by a Howard Hughes Medical Institute Predoctoral Fellowship and Stanford Graduate Fellowship.

Keratocytes were isolated from the scales of the central American cichlid *Hypsophrys nicaraguensis* and cultured as described previously^{31}. Briefly, cells were cultured in Leibovitz's L-15 medium (Gibco BRL, Invitrogen) supplemented with 14.2 mM HEPES pH 7.4, 10% fetal bovine serum (FBS) and 1% antibiotic-antimycotic (Gibco BRL), and used 1 day after isolation. 5K methoxypolyethyleneglycol 655QDs or 565QDs (Qtracker), fluorescein-conjugated 500K dextran, Texas red-conjugated 3K dextran, AlexaFluor488 free dye, AlexaFluor594 free dye (all from Molecular Probes, Invitrogen), were introduced into live keratocytes using a small volume electroporator for adherent cells^{31}. Cells were placed in a 35 mm dish in 1 ml of culture medium, and the fluorescent probe in 20 μl water was placed directly onto the cell sample. 655QDs or 565QDs were used at 2 μM for distribution measurements and at about 2 nM for single-particle tracking. Following electroporation, cells were allowed to recover in culture medium for at least 10 min. To obtain single isolated cells, sheets of keratocytes were disaggregated by incubation for 5 min in 85% PBS/2.5 mM EGTA pH 7.4 and then returned to culture medium. For visualizing the actin meshwork, cells were fixed in 4% formaldehyde and stained with TRITC-phalloidin (Molecular Probes). Blebbistatin (active enantiomer, Toronto Research Chemicals) treatment was performed at a final concentration of 50 μM blebbistatin in normal medium. Cells were imaged 10–60 min after treatment. Because of the phototoxic effects of blebbistatin on illumination with wavelength <500 nm^{32}, all pre-acquisition adjustments and focusing in the confocal ratio imaging were performed with longer wavelength illumination. No significant effects on cell morphology or speed were observed >60 s after image acquisition. All single-particle imaging was performed using longer wavelengths.

Single-particle imaging of 655QDs in motile keratocytes (~1–20 QDs/cell) was performed in a live-cell chamber at room temperature on either one of two inverted microscopes. In the first set up, cells were imaged using a ×60 (NA = 1.45) objective (1 pixel = 0.11 μm). Movies of 500–1000 frames were collected on an EMCCD (Andor IXON; Andor technology) using either an exposure and frame interval of 0.015 s or an exposure of 0.008 s and a frame interval of 0.038 s. Alternatively, cells were imaged on a Nikon Diaphot 300 microscope using a ×40 (NA = 1.3) lens. Images were collected on a cooled back-thinned CCD camera (MicroMax 512BFT; Princeton Instruments), with a ×2 optovar attached (1 pixel = 0.17 μm). Phase-contrast images were acquired before and after acquisition of a fluorescence movie of 200–500 frames in the streaming mode at a frame interval of about 0.15 s, using exposure times of 0.05 s.

Ratio imaging of the distribution of small and large probe in live motile keratocytes was performed at room temperature in a Leica SP2 AOBS confocal laser scanning microscope with a ×63 oil lens (NA = 1.4). Two probes, one green (565QDs, fluorescein-conjugated 500kD dextran or AlexaFluor488 free dye) and one red (655QDs, Texas red conjugated 3kD dextran or AlexaFluor594 free dye), were excited and imaged simultaneously using acousto-optical beam splitters that spectrally split the emitted light onto two separate detectors.

Images of fixed cells were collected with Zeiss Axioplan 2 using a ×63 oil lens (NA = 1.4). Images were collected on a cooled CCD camera (MicroMax 512BFT; Princeton Instruments), with a ×2 optovar attached.

Single-particle tracking was performed using custom-written code in Matlab version 7.0 and the image analysis toolbox (The MathWorks). A Gaussian filter was applied to the raw fluorescence images and spots were detected by thresholding. Spot positions were calculated as the centre of mass of the thresholded region, with an accuracy of about 1 pixel. This was sufficient because the particle movement during the exposure time was larger. Spots from consecutive images were linked as described previously^{33}. Tracks were bridged over short QD blinking events (<10 frames). Longer blinking events lead to the genesis of new tracks.

Measurements were performed using custom-written code in Matlab version 7.0 and the image analysis toolbox. The background-corrected intensity profiles of a pair of confocal images (red and green probe) acquired simultaneously were averaged along identical 2-μm wide (= 17 pixels) cross-sections defined manually across the central lamellipodium. A linear fit was performed to the log of the ratio between the intensities *log(n(x))=log(n _{0})-x/L* for

QD size was determined by dynamic light scattering at 20°C at a concentration of about 0.05 μM QDs in water on a DynaPro 801 instrument (Protein Solutions). The diameter of the 655QDs was measured to be 30.5 ± 1 nm, and the diameter of the 565QDs was 23 ± 2 nm.

The system of partial differential equations for the 2D hydrostatic pressure and fluid velocity distributions was analysed as described in the Supplementary Information and solved using the Virtual Cell (http://vcell.org) biological modelling framework^{34}. The corresponding 2D model is publicly available in the Virtual Cell.

Note: Supplementary Information is available on the Nature Cell Biology website.

COMPETING INTERESTS

The authors declare that they have no competing financial interest.

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