Hydrodynamic property of the cytoplasm is sufficient to mediate cytoplasmic streaming in the Caenorhabiditis elegans embryo

Abstract

Cytoplasmic streaming is a type of intracellular transport widely seen in nature. Cytoplasmic streaming in Caenorhabditis elegans at the one-cell stage is bidirectional; the flow near the cortex (“cortical flow”) is oriented toward the anterior, whereas the flow in the central region (“cytoplasmic flow”) is oriented toward the posterior. Both cortical flow and cytoplasmic flow depend on non-muscle-myosin II (NMY-2), which primarily localizes in the cortex. The manner in which NMY-2 proteins drive cytoplasmic flow in the opposite direction from remote locations has not been fully understood. In this study, we demonstrated that the hydrodynamic properties of the cytoplasm are sufficient to mediate the forces generated by the cortical myosin to drive bidirectional streaming throughout the cytoplasm. We quantified the flow velocities of cytoplasmic streaming using particle image velocimetry (PIV) and conducted a three-dimensional hydrodynamic simulation using the moving particle semiimplicit method. Our simulation quantitatively reconstructed the quantified flow velocity distribution resolved through PIV analysis. Furthermore, our PIV analyses detected microtubule-dependent flows during the pronuclear migration stage. These flows were reproduced via hydrodynamic interactions between moving pronuclei and the cytoplasm. The agreement of flow dynamics in vivo and in simulation indicates that the hydrodynamic properties of the cytoplasm are sufficient to mediate cytoplasmic streaming in C. elegans embryos.

Inside cells, various macromolecules and organelles are transported directionally to function at the appropriate locations (1, 2). Active mechanisms for directional transport involve cargo transport by motor proteins such as kinesins, dyneins, and myosins, contributing to many cellular processes (3). Cytoplasmic streaming is another prominent mechanism for active transport inside cells. Cytoplasmic streaming in the green algae Chara and Nitella has been proposed to have a role in mixing nutrients (4). In migrating cells, cytoplasmic streaming is considered to facilitate the transport of actin monomers in the direction of cell movement (5, 6).

The one-cell stage embryo of the nematode Caenorhabditis elegans is a well known system involving cytoplasmic streaming. The direction of streaming is strictly defined by cell polarity (7). The flow near the cell cortex, called cortical flow, is anterior-directed, whereas the flow distal from the cortex, called cytoplasmic flow, is posterior-directed (7, 8). Cortical flow transports proteins that are essential for cell polarity, such as PAR-3, PAR-6, and PKC-3, toward the anterior and thus contributes to establishment of the anterior-posterior polarity of the embryo (9, 10). Cytoplasmic flow transports various components toward the posterior region, and thus, the flow potentially establishes asymmetry inside the cell. However, there has been no experimental evidence so far to demonstrate such a role for cytoplasmic flow. For example, P-granules are transported along the cytoplasm (8), but it was demonstrated that the flow does not contribute to asymmetric localization of P-granules (11). Cytoplasmic flow might be important for the asymmetric localization of other components that localize posteriorly at this stage, or may stimulate association of cytoplasmic material with the posterior cortex (8, 12).

Nonmuscle myosin II (NMY-2) is required for both cortical and cytoplasmic flow (13, 14). The mechanism of cortical flow has been extensively studied (15); in contrast, the precise mechanism of cytoplasmic flow remains unknown. Importantly, NMY-2 accumulates almost exclusively on the cell cortex and the NMY-2 foci migrate toward the anterior (9). Therefore, posterior-directed cytoplasmic flow is unlikely to be driven directly by NMY-2 in the cytoplasmic region. We hypothesized that shear forces generated by NMY-2 at the cortex may be transmitted toward the central cytoplasmic region due to the hydrodynamic properties of the cytoplasm. We expected that if the cytoplasm behaves as an incompressible fluid confined in a container, the anterior-directed cortical shear force would generate countercurrent (posterior-directed) flow in the central region of the cell. In characean algae, while streaming occurs throughout the cell, the driving force for streaming is generated by myosin XI in a layer just beneath the cell surface (4, 16–23). For characean algae, a hydrodynamic simulation assuming surface myosin as the sole active force generator demonstrated that the force transmitted toward the central cytoplasmic region due to the viscosity of the cytoplasm quantitatively accounts for cytoplasmic streaming (4). An important difference between streaming in C. elegans and that in characean algae is that, in the C. elegans embryo, intense flow occurs in the central cytoplasm, the speed of which is comparable to that of cortical flow. We hypothesized that the intense flow in the myosin-less region could be explained by the hydrodynamic properties of the cytoplasm.

In this study, we determined whether our hypothesis could explain the quantitative dynamics of cytoplasmic flow in the C. elegans embryo. We tested whether we could reconstruct cytoplasmic streaming in the C. elegans embryo in vivo using a three-dimensional computational fluid dynamics simulation. We quantified the flow velocity distribution in embryos and combined these data with a hydrodynamic simulation using the moving particle semiimplicit (MPS) method. The flow velocity distribution of cytoplasmic streaming was successfully reconstructed by the fluid dynamics simulation, supporting the hypothesis that cytoplasmic flow is a fluid flow driven by cortical flow. We also demonstrated the existence of flow depending on the movements of two pronuclei rather than NMY-2, and reconstructed its dynamics using an MPS simulation.

Results

Velocity Distribution for Cytoplasmic Streaming.

We first quantified the velocity distribution for in vivo cytoplasmic streaming in the wild-type C. elegans one-cell stage embryo. The velocity distribution throughout the cytoplasm was required to compare the velocity distribution in vivo with that of the simulation. In addition, the in vivo velocity distribution at the cell cortex was required as the velocity boundary condition for the hydrodynamic simulation. Previous manual tracking had characterized the speed of cortical flow as approximately 0.09 μm/s and that of cytoplasmic flow as about 0.07 μm/s (7, 8).

To acquire the velocity distributions of the flows with high spatial resolution, we conducted particle image velocimetry (PIV) analysis (11, 15, 24, 25). We used VIT-2 ∷ GFP (26) to visualize the yolk granules, conventional probes for cytoplasmic streaming (Fig. 1A, Movie S1). In this study, we focused on the dynamics when the flows were fastest, 5–7 min before the male and female pronuclei meet in the one-cell stage embryo. We obtained the two-dimensional velocity distribution of the yolk granules at a spatial resolution of about 2 μm (Fig. 1B, upper box). To clarify the component of myosin-dependent flow, we investigated the flow velocity distribution in embryos with myosin (nmy-2) knocked down by RNAi (Fig. 1B, lower box). In the nmy-2(RNAi) embryos, the flow velocity was significantly reduced (Fig. 1C), indicating that the velocity distribution obtained for the wild-type embryos reflected NMY-2-dependent flows, as previously reported (9). Thus, we were able to acquire a high-resolution NMY-2-dependent flow velocity distribution to compare with the simulation dynamics.

Fig. 1.

PIV analysis of the flow velocity distribution in the C. elegans embryo. (A) Both cortical flow and cytoplasmic flow of GFP-labeled yolk granules were observed in embryos of the RT130 strain. We observed migration of yolk granules toward the posterior (P) pole in the central region (left) and toward the anterior (A) pole near the periphery of the cell (right). The black dots denote the position of a single granule over time. t (s) is the time after beginning observation. The black line indicates the initial position (at t = 0) of the black dot granule. (Scale bar, 10 μm.) (B) The flow velocity distribution determined by PIV represented as color maps [representative records: upper, a wild-type embryo; lower, an nmy-2(RNAi) embryo]. Red and blue denote posterior-directed and anterior-directed flow, respectively. The outline of the cell is shown as a black line. (Scale bar, 5 μm.) (C) Quantification of the flow velocity component along the AP axis (//AP) in the posterior half for wild-type and nmy-2(RNAi) embryos. Positive and negative values correspond to posterior-directed and anterior-directed velocities, respectively. White bars, flow velocities in the cytoplasmic region; gray bars, flow velocities in the cortical region. Data are means ± standard deviations (SD). Control (“WT”): n = 6 embryos; nmy-2(RNAi): n = 6 embryos. (D) The definition of x, y coordinates used in this study. x-axis is parallel to the anterior-posterior axis, and y-axis is the axis perpendicular to x-axis on the confocal plane. x = 0 at the posterior pole. y = 0 at the cortex of the middle of the cell, and maximum (y =  ∼ 13 μm) at the cell center. (E) The velocity components at the cell periphery parallel to the cell cortex (//cortex) at position x were plotted (dots). Positive and negative values correspond to posterior-directed and anterior-directed velocities, respectively. Data were fitted with a 3-segment connected line using the least-squares method.

The C. elegans embryo at the one-cell stage displays a prominent constriction known as a pseudocleavage furrow (7). At the time point when the cytoplasmic flow is most vigorous (5–7 min before pronuclear meeting), the ingression of the pseudocleavage furrow is not very obvious. The pseudocleavage furrow ingression became prominent at a later stage, about 0–4 min before pronuclear meeting. During the latter period, the cytoplasmic flow was weaker, and we could not find qualitative differences in the direction of flow with cleavage furrow formation (Fig. S1). We concluded that the pseudocleavage does not have crucial effect on the flow distribution of cytoplasmic streaming, at least when the streaming is most vigorous, which is the period we focus on in this study.

To acquire the boundary conditions for the MPS simulation, we determined the velocity at the cell periphery just beneath the cortex. The velocity component parallel to the cell cortex was plotted against the distance (x) from the posterior pole along the long axis (AP axis) (Fig. 1 D and E). To obtain the characteristics of the velocity distribution, we fitted the plot with three connected lines using the least-squares method (Fig. 1E). At x = 0 μm at the posterior pole, the speed was nearly 0. The speed increased toward the maximum value (∼0.12 μm/s) at about x = 7 μm. The speed decreased from the maximum almost linearly as x increased, reaching a minimum speed of about 0.02 μm/s at about x = 30 μm. At further anterior locations (x >  ∼ 30 μm), the speed remained low. Thus, the PIV analysis provided a velocity distribution with high spatial resolution including the boundary conditions for the hydrodynamic simulation.

Simulation of Cytoplasmic Streaming in Chara corallina.

To test whether hydrodynamics driven by cortical flow quantitatively explained the dynamics of cytoplasmic flow, we developed a hydrodynamic simulation program. As the simulation method for hydrodynamics, we chose one of the particle methods, the MPS method. The MPS method is widely used to simulate flow accompanied by moving boundaries (27). MPS is a potentially powerful method to simulate fluid dynamics inside the cell because MPS allows analysis of fluid dynamics under deforming boundary conditions, such as movements of intracellular organelles and an elastic membrane, which are difficult to be solved by conventional lattice methods.

In the MPS method, the Navier-Stokes equation Dv/Dt = μ/ρ∇²v - 1/ρ∇p is solved, where v is flow velocity, t is time, μ is viscosity ρ is mass density, and p is pressure. The Reynolds’ number is an indication of the contribution of inertial force over the viscous force (28). The Reynolds’ number of the cytoplasmic streaming of C. elegans is calculated to be approximately 10^-9. This rather low number indicates that viscosity dominates and inertia can be neglected (4, 29). In fact, the inertial term was negligibly small in our calculation. We did not actually neglect the inertial term because this would not simplify the calculation for an MPS simulation.

To obtain numerical stability, we modified the MPS method for solving the viscosity term of the Navier-Stokes equation (μ/ρ∇²v) from the conventional explicit method to the implicit method. The conventional MPS method uses the explicit method to calculate the viscosity term. In that case, a numerical stability condition is given by , where Δt is the time interval between two time steps in the simulation, and l₀ is the distance between adjacent particles at the initial time step (30). For simulation of cytoplasmic streaming in the C. elegans embryo (μ/ρ =  ∼ 0.001, Table 1), l₀ was set at ∼1 μm, at which Δt should be <  ∼ 10^-9 s to satisfy the above condition. This Δt was too small for simulating cytoplasmic streaming lasting several minutes at the one-cell stage (8). Therefore, the implicit method was used to calculate the viscosity term, because this numerical stability condition for the explicit method does not need to be fulfilled in the implicit method.

Table 1.

Parameters for simulations

Parameters		Value*	Reference
Viscosity (Ns/m2)	μ	1.0 (Ce, CF, PF) 0.001 (Cc)	Ce (35) Cc (36)
Mass density (kg/m3)	ρ	1,000	The mass density of H2O
Dimension	d	3 (Ce, Cc) 2 (CF, PF)
System size (μm) †		26 (Ce) 710 (Cc) 25 (CF, PF)
Particle density (particles/μm)		30/26 = 1.15 (Ce) 25/710 = 0.035 (Cc) 30/25 = 1.2 (CF, PF)
Distance between the adjacent particles at the initial condition (μm)	l0	0.867 (Ce) 28.4 (Cc) 0.833 (CF, PF)
Initial particle density (viscosity term) ‡	n0	14.42 (Ce, Cc) 6.540 (CF, PF)	(30)
Initial particle density (pressure term) ‡	n1	124.0 (Ce, Cc) 35.61 (CF, PF)	(30)
Laplacian coefficient	λ	7.696 × 10-13 (Ce) 8.258 × 10-10 (Cc) 4.970 × 10-13 (CF) 2.485 × 10-12 (PF)	(30)
Relative compression rate	α	10-7 (Ce, Cc, CF, PF) 10-4 (Ce-M) §	(30)

^*Ce: C. elegans, Cc: Chara Corallina, CF: Couette flow, PF: Poiseuille flow.

^†The system size for Ce is the short diameter of the cell. The system sizes for Cc, CF, and PF refer to the diameter.

^‡n₀ and n₁ are given by and , respectively, at the initial particle distribution on the square lattice. When the particle i is sufficiently inside the fluid, n₀ or n₁ reaches its maximum value.

^§Ce-M indicates simulation of microtubule-dependent cytoplasmic streaming in C. elegans (Fig. 4). See SI Materials and Methods for the details of setting this parameter.

Because previous examples of the implicit method calculation are scarce, benchmark tests were conducted. As a benchmark test, we determined whether our modified MPS approach using the implicit method to calculate the viscosity term was able to reconstruct well characterized cytoplasmic streaming (vacuole streaming) in characean algae (Fig. 2A) (4). For streaming in the algae, the cross-sectional velocity distribution has been characterized both through experimental means and numerical simulation (4, 19, 20). Therefore, simulation of this flow is an appropriate benchmark test. For our simulation, we set the velocity boundary conditions as in Goldstein et al., except that we did not include the helicity of the flow for simplicity (4). The cross-sectional velocity distribution produced by our MPS simulation (Fig. 2B, magenta) reconstructed the sigmoidal velocity distribution observed in vivo and in the previous simulation Fig. 2B, green) (4). This result indicated that the MPS simulation using the implicit method for the viscosity term calculation is able to reconstruct three-dimensional flow dynamics faithfully in the cell. We also confirmed that our MPS simulation closely reconstructed the analytical (theoretical) velocity distributions for two-dimensional Couette flow and Poiseuille flow under same viscosity (μ) and system size as in the C. elegans embryo, for which the Reynolds number is about 10^-9 (Fig. S2 A and B).

Fig. 2.

Reconstruction of cytoplasmic flow in Chara with MPS simulation. (A) Schematic representation of the simulation. We moved the wall particles surrounding the cytoplasm in two directions (right box, red and blue regions). The left circle shows the cross-sectional velocity distribution of the resultant cytoplasmic flow in the simulation. The color denotes V_x/V_max, where V_x is the velocity component in the direction of the x-axis, and V_max is the maximum velocity V_max. (B) Our MPS simulation (magenta) agreed closely with the flow velocity distribution derived from hydrodynamic simulation of Chara streaming in Goldstein et al. (green) for the region enclosed by a rectangle in (A, left). The horizontal axis shows y/R, where R is the radius of the cell. The vertical axis shows V_x/V_max.

Simulation of NMY-2-Dependent Cytoplasmic Streaming in the C. elegans Embryo.

MPS simulation was then used to determine whether the quantitative in vivo dynamics of cytoplasmic flow in the C. elegans embryo were consistent with the hypothesis that the fluid dynamics was driven by cortical flow. In our simulation, we regarded the shape of the embryo as a cylinder connected to hemispheres on both ends (Fig. 3A). We set the length of the AP axis to 55 μm, and that of the short axis to 26 μm, based on the size of the actual embryo. We seeded ∼38,000 inside particles and ∼30,000 outside particles for the cytoplasm and the cell wall, respectively.

Fig. 3.

MPS simulation of cytoplasmic flow in the C. elegans one-cell stage embryo (∼5–7 min before pronuclear meeting). (A) A snapshot of the velocity distribution in the equatorial section of the C. elegans embryo cell during the simulation (cytoplasm particles only). The cell shape was a cylinder with a hemisphere at each end. Red and blue particles indicate the particles moved into the posterior and anterior directions, respectively, in the simulation. The velocities of the green particles are near zero. (B) The velocity distribution along the AP axis in vivo reconstructed with MPS simulation. The velocity components along the AP axis (//AP) of the cortical flow and cytoplasmic flow were plotted against the distance from the posterior pole along the AP axis (x-axis). Positive and negative values correspond to posterior-directed and anterior-directed velocities, respectively. Boxes show the experimental records (gray: cortical flow, white: cytoplasmic flow). The simulation data are shown with lines (gray: cortical flow, black: cytoplasmic flow). (C) The velocity distribution along the axis perpendicular to the AP axis reconstructed with MPS simulation. The velocity components along the AP axis (//AP) in the posterior region and the anterior region were compared. Positive and negative values correspond to posterior-directed and anterior-directed velocities, respectively. Boxes show the experimental records (gray: anterior region, white: posterior region). The simulation data are shown with lines (gray: anterior region, black: posterior region).

We then set the velocity boundary conditions for the simulation. The velocity boundary conditions correspond to the velocity distribution of the outermost inside particles (in other words, “most cortical cytoplasm particles”). We set this outermost velocity distribution to reproduce the cortical velocity distribution measured in vivo (Fig. 1E). In the simulation, the movement of the inside (cytoplasm) particles was generated by moving the outside (wall) particles with “virtual” velocities. The wall particles with virtual velocities did not actually move in the simulation; however, as a result of the velocities assigned to the wall particles, they caused neighboring cytoplasmic particles to move through the viscosity interactions between the particles. In summary, by assigning virtual velocities to each of the cell wall particles, in vivo-like cortical flow was generated (Fig. 3B, gray line vs. squares).

Introduction of cortical flow resulted in generation of posterior-directed cytoplasmic flow in the simulation (Fig. 3A, Movie S2). The simulated cytoplasmic flow velocity component parallel to the AP axis was quantitatively compared with that observed in vivo. First, the velocity distribution along the AP axis was compared (Fig. 3B). In our simulation, the streamwise velocity of cytoplasmic flow reached a maximum of ∼0.1 μm/s at x =  ∼ 10 μm and gradually decreased toward the anterior at x =  ∼ 30 μm. This feature was consistent with cytoplasmic flow in vivo (Fig. 3B, black line vs. open squares).

We then compared the spanwise velocity distribution along the y-axis (Fig. 1D), the short axis perpendicular to the AP axis (Fig. 3C). The location where y = 0 μm corresponded to the most cortical locations (upper and lower ends of the cell in Fig. 1D). The most central location corresponded to y =  ∼ 13 μm. The velocity distribution in the posterior region and the anterior region were compared between the simulation and in vivo. In the simulation, in the posterior region, the velocity monotonically changed from anterior-directed to posterior-directed along the y-axis (Fig. 3C, black line vs. open squares). The stagnation point between anterior- and posterior-directed flow was located at y =  ∼ 4 μm from the cortex. The stagnation point was more interior than cortical NMY-2 localization, which has been observed within 1–2 μm from the cortex (31). Both the cortical flow velocity and the cytoplasmic flow velocity in the anterior region were markedly slower than those in the posterior region (Fig. 3C, gray line vs. squares). These quantitative features of the simulation agreed well with those observed in vivo. The agreements of both the streamwise and spanwise velocity distributions with the observed flows indicate that cytoplasmic flow in the C. elegans one-cell stage embryo can be described by fluid dynamics driven by shear force generated at the cortex.

Simulation of Microtubule-Dependent Cytoplasmic Streaming.

Through PIV analysis, in wild-type embryos, NMY-2-dependent flow was observed as expected (Fig. 1B). In addition, we detected weak cytoplasmic streaming even in nmy-2(RNAi) embryos. This streaming in the nmy-2(RNAi) embryos was qualitatively different from NMY-2-dependent streaming in the wild-type embryos in terms of timing and flow direction. The flow in the nmy-2(RNAi) embryos was fastest (∼0.02 μm/s) just before the pronuclear meeting (Fig. 4 A–C), but undetectable 5–7 min before the meeting when nmy-2-dependent strong streaming was fastest (Fig. 1B, note that the scale for velocity is different between Fig. 1B and Fig. 4B). The flow in the central region (the region with the large y value in Fig. 1D) was center-directed whereas the flow in the cortical region (the region with low y values in Fig. 1D) was pole-directed (Fig. 4C).

Fig. 4.

PIV analysis and MPS simulation of nmy-2-independent streaming dynamics. (A) Schematic representation of migration of the two pronuclei toward the center (f, female pronucleus; m, male pronucleus). (B, C) The flow in nmy-2(RNAi) embryos is dependent on microtubules. (B) The flow dynamics in nmy-2(RNAi) embryos. Weak but reproducible flow was observed in nmy-2(RNAi) embryos less than 1 min before pronuclear meeting. In nmy-2(RNAi) embryos treated with nocodazole, reproducible flow was not observed. (Scale bar, 5 μm.) (C) The flow velocities along the AP axis (//AP) in the posterior region of the cell are significantly reduced by nocodazole treatment. Data are means ± SD, measured at the timing when the flow was the most vigorous in each embryo [i.e., control (“WT”): ∼5–7 min before pronuclear meeting, nmy-2(RNAi): ∼1 min before pronuclear meeting]. WT: n = 6 embryos; nmy-2(RNAi): n = 6 embryos; nocodazole-treated nmy-2(RNAi): n = 7 embryos. (D) Velocities along the AP axis in the MPS simulation. Particles representing pronuclei (pronuclei particles) are shown inside the circle. Forces are applied to the pronuclei particles to move toward the central region. In the central region, the flow of cytoplasm particles was toward the center based on the movements of the pronuclei, whereas particles moved away from the center near the cortex.

We hypothesized that pronuclear migration (Fig. 4A) was the driving force of the streaming observed in the nmy-2(RNAi) embryos, because the timing and direction of the migration agree with those of the streaming. The flow of the cytoplasm in the nmy-2(RNAi) embryos was most prominent when pronuclei speeds become highest just before pronuclear meeting. Pronuclear migration is known to depend on the microtubule cytoskeleton (32). When we suppressed pronuclear migration by applying the microtubule polymerization inhibitor nocodazole (32), flow in the nmy-2(RNAi) embryos was impaired to a significantly weaker level (Fig. 4 B and C). Thus, flow in the nmy-2(RNAi) embryos was microtubule-dependent, supporting our hypothesis that pronuclear migration drives streaming of the cytoplasm in these embryos.

We further evaluated whether pronuclear migration was sufficient to generate microtubule-dependent flow using MPS simulation. In this paper, we have proposed that NMY-2-dependent cytoplasmic streaming is caused by countercurrent flow driven by shear force via cortical myosin, with the cytoplasm behaving as a viscous fluid that obeys the Navier-Stokes equation. If this mechanism is valid, microtubule-dependent streaming should be able to be explained using the same framework. In the simulation for microtubule-dependent streaming, we moved two pronuclei with ∼5-μm radii from near the anterior and posterior poles, respectively, toward the cell center (Fig. 4 A and D). Importantly, center-directed flow in the inner cytoplasmic region and pole-directed flow in the cortical region were observed in the simulation, as in vivo in the nmy-2(RNAi) embryos. The speed of flow was also consistent between the simulation and the experiment, as the speeds of both center-directed and pole-directed flows were within an order of magnitude of that of pronuclear migration. We did not perform further quantitative comparison because the speeds of the flows varied among individuals and locations within the cytoplasm in vivo. This variability in vivo may be due to high variability in the directions and speeds of pronuclear migration.

Discussion

In this study, we were able to reconstruct both NMY-2-dependent and microtubule-dependent cytoplasmic streaming using hydrodynamic simulations. Our results suggest that cytoplasmic flow can be regarded as hydrodynamic motion of the cytoplasm driven by actin-myosin on a thin layer near the cell cortex.

Our study provides insights into the physical properties of the cytoplasm. Previous studies have demonstrated that the cytoplasm possesses non-Newtonian properties (33, 34). If non-Newtonian properties were dominant, which is the case for cytoplasmic flow in a slime mold (34), cytoplasmic flow could not be described using the Navier-Stokes equation with constant viscosity. In this study, our MPS simulation based on Navier-Stokes equation with constant viscosity accurately reproduced cytoplasmic streaming in the C. elegans embryo. These results indicate that a Newtonian fluid is a good approximation and thus, non-Newtonian properties appear to play a minor role in cytoplasmic streaming in C. elegans. This finding is consistent with a previous microrheology study showing that simple diffusion was observed in the cytoplasm of the C. elegans embryo (35). Although non-Newtonian properties may appear when much larger or smaller forces are applied, the cytoplasm of the C. elegans embryo appears to behave as a Newtonian fluid under physiologically relevant amplitudes of applied force. Many other biological flows may also be well described as Newtonian fluids.

In this study, we assumed that cortical myosin generates a shear force against the cytoplasm to drive cytoplasmic streaming, as considered for characean algae (36). If this assumption were correct, the speed of the anterior-directed cortical myosin should be equal to that of materials just beneath the cell cortex, and this is the case for streaming in the C. elegans embryo (9). We can roughly estimate the total magnitude of cortical shear force required to drive cytoplasmic streaming. From Newton’s low of viscosity, the force per unit area required for flow is given by F_pa = μ × dv/dr, where μ is the fluid viscosity and dv/dr is the velocity gradient perpendicular to the direction of shear. Here, for simplicity, we focused on the region x = 10–20 μm from the posterior pole, where the flow was vigorous. The velocity gradient in this region was about 0.02 s^-1 (Fig. 3C). Because μ was estimated to be 1 Ns/m² for the C. elegans embryo (35), F_pa was calculated to be 0.02 N/m². The surface area of the region evaluated was about 800 μm², as the region was approximately a cylindrical shape with a radius of 13 μm. Accordingly, the total force required for streaming was estimated to be 20 pN. This estimated force acting inside the cell is of biologically relevant magnitude (37).

The biological role of cytoplasmic streaming can be postulated based on the PIV data. Whereas cortical myosin appears to strongly localize within just 1–2 μm from the cell cortex (31), our measurements showed that materials within 4 μm of the cell cortex migrated in the anterior direction (Fig. 3C). This broader cortical flow region may facilitate anterior-directed transport of the cortical PAR-3/PAR-6/PKC-3 complex, a protein complex defining the anterior cortical domain, even when it detaches from the cortex.

The results of this study demonstrated the applicability of MPS simulation to cytoplasmic streaming. Adoption of the implicit method for calculating the viscosity term is essential for systems with very low Reynolds numbers. Application of the MPS method to cytoplasmic streaming for other systems would also require use of the implicit method for viscosity term calculation. As demonstrated by the simulation of flow caused by moving pronuclei (Fig. 4), MPS has the advantage that it can easily simulate flows under conditions of moving and deforming boundaries. Because deforming boundaries are universal in biological systems, MPS is expected to have wide application for biological systems such as blood and food streaming (29, 38, 39). Our study demonstrated that the combination of flow quantification using PIV and fluid dynamics simulation using MPS is powerful in characterizing flows inside the cell. Future studies using a combination of these methods are expected to uncover further roles of streaming in vivo.

Materials and Methods

Worm Strains and RNA Interference.

C. elegans strain, RT130 (VIT-2 ∷ GFP) was maintained using standard techniques (40), and was used in this study. RNAi knockdown of nmy-2 was performed by injecting double-stranded RNA (dsRNA) as described previously (41, 42) using the following primers to amplify template DNA for dsRNA synthesis from C. elegans genomic DNA: AATTAACCCTCACTAAAGGGAGGAACGACGAAGAACTCG and TAATACGACTCACTATAGGAAGTGCGTCATCTCTGGCTT (43).

Imaging.

Live-cell imaging was conducted at room temperature using the CSU-10 Yokogawa spinning-disk confocal system as described previously (41, 42). Two-dimensional images were obtained at 2-s intervals. For nocodazole treatment, nocodazole (Sigma) was added to M9 buffer at a concentration of 25 μg/mL just before observation.

Image Analysis.

PIV was conducted with a highly accurate subpixel analysis using the gradient method (24, 44). Details of the procedure are described in SI Materials and Methods.

MPS Simulation.

MPS simulation involving solution of the Navier-Stokes equation is normally conducted using the conventional simulation algorithm using the explicit method for calculating the viscosity term (30). In this study, we used the implicit method to obtain numerical stability for low Reynolds number flow. Details of the procedure are described in SI Materials and Methods.