Live-cell imaging under centrifugation characterized the cellular force for nuclear centration in the Caenorhabditis elegans embryo

Authors: Makoto Goda, Michael Shribak, Zenki Ikeda, Naobumi Okada, Tomomi Tani, Gohta Goshima, Rudolf Oldenbourg, Akatsuki Kimura

Abstract

Our understanding of the cellular forces for nuclear positioning in cells is limited, with discrepancies existing between theoretical and empirical estimates and between species. In this study, we developed a method for the accurate measurement of these forces that combines a centrifuge polarizing microscope and orientation-independent differential interference contrast microscopy to apply controllable centrifugal forces and estimate mass densities. We applied this method to Caenorhabditis elegans embryos, revealing that the force required to move the nucleus toward the cell center was ~12 pN/μm and the frictional coefficient of the nucleus in the cytoplasm was ~980 pN s/μm.

The cell interior is highly crowded (1). Our understanding of the amount of force required to move a large object inside a crowded cell is limited (2), and the molecular mechanisms that produce the forces are unclear. The nucleus is the largest organelle in animal cells. In some cases, such as pronuclear migration after fertilization, the nucleus moves a long distance (3). Upon fertilization, the sperm-derived pronucleus forms at a peripheral position where the sperm enters the oocyte. Its migration to the center of the fertilized egg is mainly driven by microtubule asters (4–6). Microtubules can generate force by dynamically elongating and shrinking or by acting as a rail for molecular motors (7). They often extend radially from the centrosome, a microtubule organizing center (MTOC), forming an aster. Since the microtubule aster itself is a large cellular structure, the migration of the pronucleus together with the asters in the crowded cytoplasm is expected to require large forces (8).

The forces generated in the cell to move the pronucleus during its migration were first characterized in fertilized sea urchin eggs (9). Tanimoto et al. used magnetic tweezers to measure the force required to move the complex consisting of the microtubule asters (extending up to ~50 μm to reach the cell cortex), sperm-derived pronucleus, and oocyte-derived pronucleus. They revealed that the microtubule aster produces a force of 580 ± 21 pN to move the nuclei–aster complex, whose frictional coefficient is 8,400 ± 280 pN s/μm. Pulling of the microtubules and thus the nuclei–aster complex at the cytoplasm by cytoplasmic dynein is thought to generate the migration force in sea urchin (9–11) and in other organisms (6, 12, 13). However, another report argues that dynein is not required for the pronuclear migration in sea urchins (14). Thus, the molecular mechanism underlying force production and the high frictional coefficient of the nuclei–aster complex in sea urchin eggs remain unclear.

The nematode Caenorhabditis elegans is another popular model owing to the ease of live imaging and gene manipulation. The forces generated by the microtubule aster and the frictional coefficient to move the nuclei–aster complex for the pronuclear centration have not been measured experimentally in this system. In theory, the frictional coefficient can be estimated as the frictional coefficient, Fdrag/V, of a spherical object inside a simple viscous fluid (i.e., Newtonian fluid) follows Stokes’ law (15), as Fdrag/V = 6πηR, where Fdrag is the force for dragging the sphere, V is the velocity of the sphere movement, η is the viscosity of the medium, and R is the radius of the sphere. In C. elegans, the sperm-derived pronucleus is a spherical object with a radius of 4.5 μm and an estimated viscosity of the cytoplasm is 0.2 pN s/μm^2^ (16). Applying these values to Stokes’ equation, the frictional coefficient is ~20 pN s/μm. The movement of the nucleus has a compressive effect on the cytoplasm, and simulation studies have shown that the effect of confinement increases the coefficient by 3.3-fold (8, 17). Furthermore, the nuclei–aster complex is not a smooth sphere; a simulation study compared it to a porous medium with a sixfold increased frictional coefficient (8). This calculation is consistent with an experimental observation that the movement of a sperm pronucleus with microtubule asters is 4.4-fold slower than that of an oocyte pronucleus without microtubule asters (18). The theoretically estimated value of the frictional coefficient of the nuclei–aster complex in the C. elegans embryo is ~300 pN s/μm, approximately 30-fold smaller than the experimentally measured value in sea urchin (8,400 pN s/μm) (9). The maximum speed of pronuclear migration in C. elegans is ~0.1 μm/s. The force required to generate such speed with the frictional coefficient of 300 pN s/μm is estimated to be 30 pN, approximately 20-fold smaller than the experimental measurement in sea urchins (580 pN). To resolve these discrepancies, experimental measurements of the force required for C. elegans pronuclear migration are needed.

Garzon-Coral et al. measured the forces required for positioning the mitotic spindle in C. elegans (16). In contrast, the measurement of forces related to nuclear positioning in the C. elegans embryo has not been reported. The centrifuge microscope (19) is a promising tool for measuring intracellular forces as it allows live imaging of a microscopic specimen under centrifugal forces. If the densities of the nucleus and cytoplasm are different, the nucleus should move depending on the centrifugal speed according to the following Fcfg = Δρ×NV×RCF×g. Fcfg is the centrifugal force acting on the nucleus, Δρ is the density difference between the nucleus and the cytoplasm, NV is the volume of the nucleus, RCF (relative centrifugal force) is the centrifugal acceleration relative to the gravitational acceleration, g (9.8 m/s^2^). Even if the density difference is small, the nucleus will be moved by the centrifugal force if RCF is high enough. The centrifuge polarizing microscope (CPM) invented by Inoué et al. is capable of imaging under a rotational speed that generates up to ~10,000×g (19, 20). The CPM has been used to apply forces to various biological samples (21–23). In this study, by applying centrifugal forces using the CPM, we measured the cellular forces and frictional coefficient for nuclear centration in C. elegans embryos.

Results

Movement of the Nucleus in the C. elegans Embryo Under Centrifugal Forces.

In the CPM (19), the rotor spins between the objective (40×, NA 0.55) and condenser lenses (Fig. 1A). The specimen in the rotor is illuminated stroboscopically by 6-ns laser pulses, which are synchronized to the exact timing when the specimen comes under the objective lens. Therefore, despite the fast rotation of the rotor, the camera image is stationary with a high resolution of up to 1 μm.

We mounted the C. elegans embryo in a chamber designed for the CPM. Inside the chamber, a density gradient of Percoll was established, and the position of a floating embryo was fixed in the chamber relative to the Percoll gradient (Fig. 1 A′ and A″). The anterior pole of the C. elegans embryo always faced the center of the rotation stage during the one-cell stage (Fig. 1 B and C). The reproducible orientation of the anterior pole to the center was caused by the anterior localization of a low-mass-density material (see the section; Calculation of the Centrifugal Force: Mass Density).

We then observed the movement of the pronuclei. Under a low centrifugal speed (500 rpm, 21×g, Fig. 1B and Movie S1), the movement of the pronuclei was similar to that in the noncentrifugation condition (for example, see figure 1 in ref. 6). The sperm- and oocyte-derived pronuclei formed at the posterior and anterior positions, respectively, and met and attached to each other at a posterior position (pronuclear meeting, time point 0 in this study). Following attachment, the pronuclei moved together toward the center of the embryo, where they were stabilized until mitosis.

In contrast, when we increased the centrifugal speed to 2,500 rpm (520×g) after the pronuclear meeting, the pronuclei passed the center of the cell and moved further toward the center of the rotation stage (Fig. 1C and Movie S2). These observations revealed that i) the pronuclei can be moved by centrifugal forces produced by the CPM, and ii) the mass density of the pronuclei is lower than that of the cytoplasm.

Even though nuclei migrated to the anterior pole, the mitotic spindle, which is formed after nuclear envelope breakdown, moved back to the normal position and divided normally (Movie S2). This is likely because the mitotic spindle is not lighter than the cytoplasm, and the centrifugal force does not act on it. Embryos subjected to centrifugation up to 520×g did not show stratification of the cytoplasm (Movies S1 and S2, n = 29), whereas centrifugation at 2,100×g (5,000 rpm) caused stratification (Movie S3, n = 11). We evaluated the effect of centrifugation on organelle distribution. We speculated that the yolk granule is one of the heaviest organelles in the embryo, based on a previous centrifugation experiment using sea urchin eggs (24). The yolk granules of the C. elegans embryo (SI Appendix, Fig. S1A) were asymmetrically distributed to the posterior cells after rotating the embryo at 2,100×g (5,000 rpm) for ~3 min before the first cell division, indicating that the yolk granule is heavier than the bulk cytoplasm (SI Appendix, Fig. S1B, n = 5). The sedimentation of the yolk granules was not observed after centrifugation at 520×g (2,500 rpm) for ~3 min (SI Appendix, Fig. S1C, n = 5), supporting that detectable changes in the distribution of the organelles other than the nuclei do not occur in this condition. In addition, embryos subjected to 520×g from pronuclear migration until the first cell division hatched normally after unmounting from the CPM chamber and culturing under normal gravity. We concluded that a centrifugal force of up to 520×g produces a mild perturbation and does not cause detectable damage to cellular structures related to pronuclear migration. We thus used this experimental setup to relate the external force provided using centrifugation and the movement of the pronuclei to quantitatively evaluate the forces produced by the cell.

Calculation of the Centrifugal Force: Nuclear Volume.

The centrifugal forces acting on an object can be calculated as Fcfg = Δρ×NV×RCF×g. To calculate the force from RCF, which is a function of the centrifuge speed, the parameters NV (nuclear volume) and Δρ (density difference between the nucleus and the cytoplasm) are necessary. We quantified NV from confocal microscopy images of cells whose nuclear membrane was fluorescently labeled (Table 1 and SI Appendix, Fig. S2). We focused on the movement of the pronuclei after the sperm- and oocyte-derived pronuclei meet. This is because a high centrifugal force applied before the two pronuclei meet can prevent the meeting, making the comparison between different RCF conditions difficult. The total volumes of the two pronuclei were 730 ± 90 μm^3^ (mean ± SD from 13 embryos at 51 timepoints).

Calculation of the Centrifugal Force: Mass Density.

To quantify Δρ, which is the density difference between thenucleus and the cytoplasm, we used an orientation-independent differential interference contrast (OI-DIC) microscope (25, 26). The OI-DIC microscope measures the optical path difference (OPD) within a specimen. From the OPD and the thickness of the object, the refractive index can be calculated. Because the refractive index is proportional to the dry mass and because the refractive increments of most substances in cells are approximately the same (27–29), mass density in the living cell can be calculated from the refractive index (30).

When we imaged the C. elegans embryo using the OI-DIC microscope, the refractive index outside the cell but inside the eggshell was comparable to that of the buffer and lower than that of the cell (Fig. 2 *A-*iii, asterisk). This area is referred to as the extra-embryonic matrix (EEM) (31). The low refractive index of the EEM concentrated on the anterior side of the embryo suggested that the mass density of the anterior side is lower than that of the posterior side. This observation is consistent with our CPM observation that the anterior pole always faces toward the center of the rotation (Fig. 1 B and C). These results further indicated that OI-DIC microscopy is a reliable method to estimate the mass density inside the cell.

The lower refractive index of the nuclear region compared with that of the cytoplasm (Fig. 2A) indicated that the mass density of the nucleus is lower than that of the cytoplasm, in agreement with the movement of the nucleus depending on the centrifugal force (Fig. 1). From the 2-dimensional map of the OPD (Fig. 2 A-i), we quantified the densities of the cytoplasm and nucleus by fitting the OI-DIC microscopy data to a formula assuming that the cross-section of the nucleus and the cell perpendicular to the long axis of the C. elegans embryo are circular (Fig. 2B; see Materials and Methods). As calculated from the OPD, thickness, and refractive index of the buffer (1.33), the density of the cytoplasm was 1,063 ± 8 mg/mL, and the difference in density between the nucleus and the cytoplasm (Δρ) was 37 ± 7 mg/mL (mean ± SD from six nuclei of five embryos).

The conversion of the OPD to the density difference of cell compartments assumed a linear relationship between the refractive index and dry mass, and a specific refractive index increment of 0.0018 [100 mL/g] (28, 30). To validate this assumption, we measured the mass density of the embryo with an independent method using the CPM. We mounted the embryo together with colored standard beads of known density. As we applied centrifugal force to the mixture, the positions of the standard beads separated along the Percoll density gradient. The positions of embryos were almost identical to those of a bead with a known density of 1.06 g/mL (Fig. 2C), which agreed with the mass density of the cytoplasm (1,063 ± 8 mg/mL) estimated using the OI-DIC microscope.

In conclusion, we adopted a density difference (Δρ) of 37 mg/mL to calculate the centrifugal force on pronuclei.

Strategy to Quantify Cellular Forces for Nuclear Centration and the Frictional Coefficient of the Nuclei–Aster Complex.

From the abovementioned measurements of the NV and Δρ, we can calculate the amount of centrifugal force applied to the nuclei (Fcfg). The purpose of this study is to quantify the force produced by the cell to move and maintain the nuclei at the center of the cell (Fcentration). In addition, the drag force (Fdrag) works to move nuclei in the opposite direction. We assume that these three forces are the major forces acting on the pronuclei and are balanced as Fcentration + Fdrag = Fcfg (Fig. 3 *A-*i). The assumption is reasonable because, without centrifugal force (Fcfg = 0), the nuclei stop (Fdrag = 0) at the center of the cell, where Fcentration should be 0.

Using this equation, we quantified Fcentration and Fdrag. When the nuclei are not moving, the drag force is zero (Fdrag = 0). In this condition, the centration force equals the centrifugation force (Fcentration = Fcfg) (Fig. 3 *A-*ii). Similarly, when the nuclei are at the center, the centration force should be zero (Fcentration = 0). In this condition, the drag force equals the centrifugation force (Fdrag = Fcfg) (Fig. 3 *A-*iii). We, therefore, focused on the position where the nuclei stop moving, Leq, to determine Fcentration and the speed of the nuclei when they pass the center of the cell, Vcen, to determine Fdrag under various centrifugation conditions.

Fcentration: The Force Produced by the Cell to Move and Maintain the Nuclei at the Cell Center.

We tracked the positions of pronuclei under various centrifugal accelerations (Fig. 1D) and fitted the trajectory after the centrifugal speed reached its target values with a formula describing the relationship between the position of the center of the pronuclei, x, as a function of time, t.

[1]x(t) = Leq × [1-exp{-(t - t0)/τ}]

where Leq is the position where the pronuclei stop moving (i.e., Fcfg = Fcentration), t0 is the time when the pronuclei pass the center, and τ is a characteristic time scale for the movement. This function is often used to represent trajectories approaching a certain point (x = Leq in this case) with decreasing speed and fits the experimental data well (red dotted line in Fig. 1D).

We detected a roughly linear relationship between the centrifugal force and Leq, i.e., how far the pronuclei were displaced from the center (Fig. 3B). This indicates that the cellular machinery to bring nuclei to the center of the cell behaves like a Hookean the further the nuclei are displaced (x), the stronger the forces generated toward the center (Fcentration = −K × x, where K is the stiffness of the centering spring). We concluded that the Hookean spring mechanism accounts for the majority of the relationship between force and distance. The imperfect fit to the linear line (the coefficient of determination, R^2^ = 0.81) may reflect experimental noise and/or imply nonlinear effects, such as the confinement effect (8). The slope of the fitted regression line (βRL in Table 1) was (2.3 ± 0.3) × 10^−2^ μm/[×g] (mean ± SD, n = 29) (Fig. 3B). Based on the NV and Δρ measured in this study, the stiffness K of the “Hookean spring” for the centration of the nuclei in the cell was K = 12 ± 3 pN/μm (mean ± SD) (Table 1). In summary, we succeeded in establishing a method to measure the nuclear centration force utilizing centrifugal forces, and in measuring the force in C. elegans embryos.

Fdrag: Drag Force.

To characterize the drag force for nuclear centration, we focused on the speed of the nuclei when they pass the center, Vcen (i.e., Fcentration = 0). The speed was calculated from fitting the trajectory after the centrifugal speed reached its target values to Eq. 1 (red dotted line in Fig. 1D) as Vcen = Leq/τ. We plotted the speed against centrifugal acceleration (Fig. 3C), revealing roughly a linear relationship. In viscosity-dominant conditions, or a low Reynolds number regime (32), the drag force (Fdrag) is proportional to the velocity of an object (V). The frictional coefficient Cfric is the ratio between them, Cfric = Fdrag/V. The intracellular environment is considered to be viscosity-dominant, and our measurements agree with this notion. We concluded that the proportional relationship largely explains the relationship between force and velocity. The imperfect fit to the linear line (the coefficient of determination, R^2^ = 0.88) may reflect experimental noise and/or imply additional mechanisms, such as the viscoelasticity of the cytoplasm (16) or the nonspherical nature of the nucleus–aster complex (8). The slope of the regression line (βRV in Table 1) was (2.7 ± 0.4) × 10^−4^ μm/s [×g] (mean ± SD, n = 29). The frictional coefficient, Cfric, was calculated as Cfric = Fdrag/V = NV × Δρ × RCF × g/V = NV × Δρ × g/βRV = 980 ± 250 pN s/μm (mean ± SD).

Ratio of Stiffness and the Frictional Coefficient Did Not Depend on the Centrifugation.

We characterized the centration force provided by the cell as the stiffness of the centering spring, K, and the nuclear drag as the frictional coefficient, Cfric. The values should not depend on the centrifugation if the centrifugation does not affect the cellular machinery for nuclear centration. To check whether the centrifugation affects the parameters K and Cfric, we focused on the ratio of K and Cfric, i.e., Cfric/K. In our model (Eq. 4), this ratio corresponds to the relaxation time, τ, of the exponential decay of the distance to the equilibrium position of the nuclei (Fig. 1D) (see Materials and Methods; The relaxation time τ as the ratio of the centering spring K and the frictional coefficient Cfric). The relaxation time τ (τ = Cfric/K) was obtained from each measurement by fitting the trajectory after the centrifugal speed reached its target values to Eq. 4 in Materials and Methods. When we plotted the τ values against centrifugal acceleration (Fig. 4A), the coefficient of determination (R^2^) was 0.08, and the F-value was 0.14, when performing linear regression analysis, indicating that the τ value does not correlate with centrifugal force.

In our model (Eq. 4), we assumed that the stiffness of the centering spring (K) and the frictional coefficient (Cfric) are constant during the nuclear migration. Here, we investigated the possibility that the stiffness and frictional coefficient change during the migration. If this is the case, the τ value may differ from Cfric/K. The stiffness and frictional coefficient may change during the migration if the remodeling of the centering machinery and the cytoplasmic friction is slower than the time-scale of the migration by, for example, a slow increase in the length of astral microtubules. We considered the remodeling is faster than the nuclear migration from the following observations. The elongation speed of the microtubule [~0.7 μm/s (33)] is faster than the migration speed of the nuclei (Fig. 1D). The nuclei were stably positioned as they reached the plateau position after the migration until nuclear envelope breakdown (Fig. 1D). The τ value was almost constant regardless of the centrifugal force and thus of the migration speed (Fig. 4A). The results supported that the stiffness of the centering spring and the frictional coefficient are almost constant during the migration, and thus the discrepancy between τ and Cfric/K is small, if any.

To further investigate the effect of centrifugation on the parameters, we compared the τ values from the centrifugation experiment with those from experiments without centrifugation. For the latter, we analyzed the trajectories of the nuclei from both DIC images of the N2 strain (white circles in Fig. 4B) and confocal images of a :GFP expressing strain (white triangles in Fig. 4B) in the noncentrifugation condition. The relaxation time τ (=Cfric/K) was obtained from each measurement by fitting the trajectory to Eq. 5′ for noncentrifugation conditions (see Materials and Methods, and SI Appendix, Fig. S3, dotted line). The values did not change significantly between the centrifugation and noncentrifugation conditions (Fig. 4B, P >0.05, Student’s t test).

In summary, the results that the τ value did not change depending on the centrifugation support the notion that the force produced by the centration machinery is not affected by our centrifugation procedure. The stiffness of the centering spring, K, and frictional coefficient of the nucleus, Cfric, obtained in this study should reflect that of the normal (noncentrifugation) condition.

Measurement of Nuclear Motion in Mutant Cells.

The force to center the pronuclei has been quantified in sea urchin eggs using magnetic tweezers (9). The current study, quantifying the force in C. elegans, adds another organism for comparison. In addition, our method provides a way to characterize gene functions for nuclear centration, as we can easily manipulate gene activity in C. elegans. As a proof of concept, we examined the nuclear movement of zyg-9 and zyg-12 mutant embryos under centrifugal forces (Fig. 5). zyg-9 encodes a Xenopus microtubule-associated protein (XMAP215) homolog that promotes the elongation of microtubules (34–36) (Fig. 5A). The zyg-12 gene encodes a functional KASH protein in C. elegans essential for the attachment of microtubule asters to both sperm and oocyte pronuclei (37) (Fig. 5A). In zyg-9 or zyg-12 temperature-sensitive mutant embryos, active pronuclear migration does not occur (34, 37, 38) at the restrictive temperature because microtubule growth and attachment to the pronucleus are required (38).

Under slow centrifugation (500 rpm, 21×g), the pronuclei showed little movement (Fig. 5B), as observed under no centrifugation. We increased the centrifugation speed for the zyg-9 (b244ts) and zyg-12 (ct350ts) mutant embryos to 2,000 rpm (340×g) at the timing of the relaxation of the pseudocleavage furrow, which corresponds to the timing of pronuclear meeting in control cells. Both the sperm and oocyte pronucleus moved centripetally at almost constant speed. We measured the speed as a function of rotational speed to calculate the frictional coefficient in zyg-9 and zyg-12 mutant embryos. The slopes of the regression line were (4.3 ± 0.4) × 10^−4^ μm/s [×g] (mean ± SD, n = 8) for the zyg-9 mutant and (3.3 ± 0.4) × 10^−4^ μm/s [×g] (mean ± SD, n = 8) for the zyg-12 mutant. These values are for one (sperm-derived) pronucleus but not for the two attached pronuclei, as measured for wild-type embryos above. As the wild-type control for the frictional coefficient of the sperm-derived pronucleus, unlike the previous experiments, we increased the centrifugation speed before the pronuclear meeting and measured the velocity of the sperm-derived pronucleus when it crossed the center of the cell. The slope of the regression line was (2.0 ± 0.6) × 10^−4^ μm/s [×g] (mean ± SD, n = 5) for the wild-type sperm-derived pronucleus. Based on the volume of the pronucleus and the density difference, the frictional coefficients for the sperm-derived pronucleus were 310 ± 80, 410 ± 10, and 650 ± 250 pN s/μm (mean ± SD) for zyg-9, zyg-12, and wild-type, respectively.

Three quantitative features should be noted here. First, the frictional coefficient was reduced to approximately half in zyg-9 and zyg-12 pronuclei, where the microtubule asters are short or detached from the pronucleus, compared to that in the wild-type. This was reasonable because the microtubule asters associated with the pronucleus should increase friction when the pronucleus moves inside the cytoplasm (8, 18). Quantitatively, the difference of about twofold was smaller than the previous theoretical estimation that the microtubule aster should increase the frictional coefficient sixfold (8). Second, for the wild-type, the frictional coefficient of the complex of one pronucleus and the aster (650 pN s/μm, “SPN only” in Table 1) was 1.5-fold smaller than that of the complex of two pronuclei and the aster (980 pN s/μm, Fig. 3C, “SPN and OPN” in Table 1). This indicated the contribution of the oocyte pronucleus to the friction. Third, the frictional coefficient for the SPN only of the wild-type (~650 pN s/μm) was higher than the theoretical estimation of ~300 pN s/μm (8, 18). Similarly, the frictional coefficients of the zyg-9 and zyg-12 mutants (~300 to 400 pN s/μm) were higher than the theoretical estimation for the nucleus without an aster (~50 pN s/μm) (8, 17). The quantitative difference between our measurement and the theoretical estimates based on Stokes’ law, the confinement effect (8, 17), and the contribution of microtubule asters (8) suggests that additional components contribute to the frictional coefficient of the pronuclei–aster complex (Discussion).

Discussion

Mass Density of the Nucleus is Lower Than that of the Cytoplasm in the C. elegans Embryo.

Our CPM observation and OI-DIC microscope measurement both indicated that the nucleus has a lower mass density compared to that in the cytoplasm in the C. elegans embryo. This might be counterintuitive, as we know that the nuclei often sediment to the heaviest layer in cell fractionation experiments by centrifuging homogenized cells. On the contrary, the low mass density of the nucleus seems to be a general feature of the living cell (i.e., nonhomogenized). The centrifugation of the unfertilized eggs of the sea urchin, Arbacia punctulate (24), or the embryo of the nematode Diploscapter pachys (39) showed that the nucleus moves to the centripetal pole and thus has a lower mass density than that of the cytoplasm. The imaging technique used in this study, similar to OI-DIC microscopy, for various cultured cells showed that the refractive index of the nucleoplasm is smaller than that of the cytoplasm, indicating that the mass density of the nucleus is lower than that of the cytoplasm (40–42). Our study conducted refractive-index imaging and centrifugal microscopy of a same type of living cells. The estimation of the mass density from the refractive index imaging (OI-DIC microscopy) was demonstrated to be accurate both qualitatively (i.e., the nucleus has a lower mass density) and quantitatively (i.e., the mass density of the cytoplasm is ~1.06 g/mL, Fig. 2), using the CPM.

Value of the Centration Force.

We established an experimental setup using a centrifugal microscope to quantify the force required for nuclear migration toward the center of the cell. The force was almost proportional to the distance of the nuclei from the center. The force per displacement required to move the nuclei was 12 ± 3 pN/μm (stiffness, K). The maximum force produced during migration was approximately 100 pN, based on the frictional coefficient (980 pN s/μm) and maximum velocity of the nuclei (~0.1 μm/s, SI Appendix, Fig. S3).

The magnitude of the force required to displace the nuclei at prophase of the one-cell stage measured with the CPM in this study (12 pN/μm) was similar to that required to displace a pole of the mitotic spindle at metaphase measured using magnetic tweezers (16 pN/μm) (16), despite the differences in cell cycle (prophase vs. metaphase) and cargos (nuclei vs. spindle). This similarity suggests that the centering of the pronuclei at prophase and the mitotic spindle at metaphase are accomplished by a common mechanism, although the mechanisms proposed in past studies differ (6, 12, 16, 43).

The maximum force produced for C. elegans pronuclear migration was approximately 1/6 of the force measured in a sea urchin (580 pN) (9). A simple explanation for this discrepancy is the difference in cell size (an ellipsoid with a long axis of 50 μm and short axes of 30 μm for the C. elegans embryo, and a sphere with a diameter of 90 μm for the sea urchin egg). If the driving forces are produced throughout the cytoplasm and thus depend on the length of the microtubule, as proposed in these organisms (6, 9–12, 44), it is reasonable for the generated force to depend on the cell size. However, the force-producing mechanism for centration is currently under debate and may involve the microtubule pushing against the cell cortex (14, 16) or the microtubule pulling from the cell cortex (43, 45).

Value of the Frictional Coefficient of the Nuclei–Aster Complex.

The frictional coefficient of the nuclei–aster complex, 980 ± 250 pN s/μm (for the two pronuclei), was approximately 1/8 of the corresponding estimate for the sea urchin (8,400 pN s/μm) (9). The difference may be explained again by the size difference. As the astral microtubules become longer in the larger sea urchin eggs, it will be harder to move the asters.

Based on past theoretical and experimental considerations using C. elegans embryos (8, 16–18), the frictional coefficient of a pronucleus with the aster was estimated to be 300 pN s/μm. The experimental measurements in the current study were approximately twofold higher (650 [pN s/μm] for a pronucleus with the aster, Fig. 5B). The difference may be related to the assumption of a Newtonian fluid in the estimate. In a simple viscous fluid, the frictional coefficient is proportional to the radius of the sphere, and the estimate applies a cytoplasmic viscosity of 0.2 pN s/μm^2^ measured using ~1 μm beads (16). In contrast, the real cytoplasm is filled with cytoskeletal filaments and organelles, where the frictional coefficient may increase nonlinearly, more than proportional to the radius of the sphere. Our measurements indicate that such obstacles for large organelles suppress the movement of the nuclei and increase the frictional coefficient. It will be an interesting future direction to search for the molecular entity of the obstacles by manipulating cytoskeletons or organelles by gene knockdowns and measuring the frictional coefficient using the CPM.

A previous theoretical study predicted that microtubule asters should increase the frictional coefficient sixfold (8). A comparison of the speeds of sperm- and oocyte-derived pronuclei suggested that the difference is 4.4-fold (18). The twofold difference in the frictional coefficient observed between zyg-9 (b244ts) and wild-type embryos in our study suggests that the effect of microtubule asters is not large, although we must note that zyg-9 (b244ts) does not completely eliminate astral microtubules. Further CPM measurements using C. elegans embryos with various mutants and gene knockdowns will help characterize the role of astral microtubules in force production and the frictional coefficient for pronuclear migration.

Centrosome-Organelle Mutual Pulling Model.

The mechanisms that produce forces for nuclear centration are still under debate. In most cases, centration depends on microtubule functions. However, it is not clear whether the force for centration is generated by the pushing of microtubules against the cell cortex (14, 16, 46) or by the pulling of microtubules by a minus-end directed motor, cytoplasmic dynein (3, 47). The pulling model is further divided into two or more models depending on whether the pulling occurs throughout the cytoplasm (6, 9–13, 48) or at the cell cortex (43, 45, 49). Pulling at the aster periphery (50) and at the interphase of the aster and actin network (51) have been proposed recently. The mechanism might differ among species.

The amount of force measured in this study is consistent with the cytoplasmic pulling model and, more specifically, the organelle-centrosome mutual pulling mechanism (3, 12, 52). According to the current study, 12 pN of the forces work on the nucleus when the nucleus is 1 μm from the center to move the nucleus toward the center. At this distance, the volume difference of the cytoplasm toward the direction of movement and in the opposite (rear) direction will be ~5% of the total volume of the embryo. In the organelle-centrosome mutual pulling model, the force toward the center is roughly proportional to the volume difference between the direction of movement and the opposite direction. This means that a force of 12 pN is generated in the 5% volume fraction of the embryo. Thus, we estimate that a total force of ~240 pN is generated in the cytoplasm of the organelle-centrosome mutual pulling model that explains nuclear centration. This is a reasonable amount. Moving single organelles requires several pN of force, indicating that approximately 100 organelles are moving at a time, which is roughly consistent with the number of moving organelles observed in vivo (12).

Combined Use of CPM and OI-DIC as a Tool to Characterize Forces within the Cell.

This study used centrifugal force to measure the forces inside the cell (53). Compared to the force measurements using magnetic or optical tweezers, this method does not require the injection of large beads into the cell, and thus the experiment can be repeated easily.

Microtubule-based structures responsible for nuclear centration likely remained intact under the centrifugation applied in this study. First, the relaxation time of the nuclear movement, which should reflect the ratio between the stiffness and frictional coefficient parameters, did not depend on the speed of centrifugation (Fig. 4A) and did not differ between the centrifuge and noncentrifuge conditions (Fig. 4B). The results supported that the nuclear centration machinery and the cellular friction did not change using centrifugation. Second, no sign of relocalization of the organelles was detected with centrifugation up to 2,500 rpm (SI Appendix, Fig. S1 and Movies S1–S3). Third, after the nuclear envelope broke down, the mitotic spindle rapidly moved back to the center of the cell, even under centrifugation (Movie S2). Because the centering of the spindle is driven by microtubule-generated forces (16), microtubule structures for centration of the mitotic spindle remain intact under centrifugation. Finally, the embryos hatched after centrifugation, meaning that the centrifugation did not disrupt cellular machinery.

OI-DIC microscopy provided a reasonable estimation of the mass density of the cell, and its combination with CPM provided a method to apply controllable forces to intracellular structures. The application is limited to nuclei currently; however, searching for appropriate biological systems and/or developing protocols to use centrifugal forces (e.g., injecting high-density beads into the cell) will provide new ways to investigate the forces acting inside living cells.

Materials and Methods

C. elegans Strains.

C. elegans strains were cultured using standard procedures (54). The N2 (wild-type), RT130 (pwIs23[vit-2::GFP]), CAL0332 (unc-119(ed3) III; ltIs38[Ppie-1::GFP::PH(PLC1delta1); unc-119(+)]; ltls37[Ppie-1::mCherry::HIS-58; unc-119(+)]), and XA3507 (unc-119(ed3) III; qals3507[unc-119(+) + Ppie-1::GFP::LEM-2] III) strains were maintained at 22 °C. The DH244 (zyg-9(b244ts) II) and BW54 (zyg-12(ct350ts) II) strains were maintained at 16 °C and shifted up to 25 °C just before the observation.

Centrifuge-Polarizing Microscope Analyses.

The CPM system developed at the Marine Biological Laboratory was used (19). C. elegans embryos were cut out of the adult worms, and embryos before the pseudocleavage stage were selected. Embryos together with 4 μL of 0.75× egg-salt buffer were layered on top of 8 μL of 0.75× egg-salt buffer with 75% Percoll (vol/vol; Sigma, St. Louis, MO) and spun in the CPM. Inside the CPM chamber, a density gradient of the Percoll was formed, and the embryo was stably positioned according to its mass density (e.g., Fig. 2C). The gradient was stably formed immediately after the start of the rotation, based on the observation that the embryo does not move after we find the embryo under the microscope (typically, less than a minute after the onset of the rotation). The distance of the chamber from the center of the rotation stage was 7.5 cm. RCF was calculated from rpm of N as RCF = N^2^ × 8.4 × 10^−5^.

In this study, we used 500 rpm as the basal condition to create the Percoll gradient in the chamber and check the status of the embryo. To examine the movement of the two pronuclei after the meeting (Figs. 1, 3, and 4, “SPN+OPN” in Table 1), the rotation speed was increased from 500 rpm to a target speed when the two pronuclei met. To examine the movement of the sperm pronucleus in zyg-9 and zyg-12 mutants (Fig. 5, SPN only in Table 1), the rotation speed was increased at the timing of the relaxation of the pseudocleavage furrow, which corresponds to the timing of pronuclear meeting in control cells. For the control experiments in wild-type, the centrifugation speed was increased before the pronuclear meeting to prevent the meeting, and the velocity of the sperm-derived pronucleus was measured when it crossed the center of the cell. To examine the distribution of the yolk granule after the first cell division (SI Appendix, Fig. S1), the rotation speed was increased 1 min after nuclear envelope breakdown, which is 3 to 4 min before cytokinesis.

Differential interference contrast images were obtained using a 40×, 0.55 N.A. objective lens (SLCPlanFI; Olympus, Tokyo, Japan) with a 10×, 0.30 N.A. condenser (UPlanFI; Olympus) with a zoom ocular set to ×1.5 (Nikon, Tokyo, Japan). The specimen was momentarily illuminated by a 532-nm wavelength, 6-ns laser pulse (New Wave Research, Fremont, CA), and images were captured by an interference-fringe-free CCD camera (modified C5948; Hamamatsu Photonics, Hamamatsu, Japan).

To measure the density of the embryo (Fig. 2C), the “Density Marker Beads Kit (1.02, 1.04, 1.06, 1.08, 1.09, 1.13 g/cc)” (Cospheric, Cat# DMB-kit, Somis, CA) was used. The buffer of the beads was exchanged for 0.75× egg-salt buffer, and several beads of each density were mounted on the CPM chamber together with C. elegans N2 embryos as described above. Images were captured at a rotation speed of 500 rpm. The beads of different densities were distinguished by color.

Time-lapse images were analyzed using ImageJ (NIH, Bethesda, MD). The coordinates of the center of the sperm- and oocyte-pronucleus, and those of the anterior and posterior poles of the cell, were quantified via manual tracking. The midpoint of the center of the sperm- and oocyte-pronucleus was defined as the center of the pronuclei. The positions of the centers of the pronuclei along the anterior–posterior (AP) axis after the pronuclear meeting were calculated and plotted against time (Fig. 1D). The plot after the centrifugal speed reached its target values was fitted to the function x = Leq × [1 − exp{−(t − t0)/τ}] using the Microsoft Excel “solver” tool, where x is the position of the centers of the pronuclei (x = 0 at the cell center and x >0 for the anterior half), Leq is the position where the pronuclei stop moving, t is the time, t0 is the time when the pronuclei pass the center, and τ is a characteristic time-scale for the movement.

Relaxation Time τ as the Ratio of the Centering Spring K and the Frictional Coefficient Cfric.

In the centrifugation experiment, nuclear movement is driven by the sum of the centering force (Fcentration) provided by the cellular machinery and the centrifugal force (Fcfg). The sum of the force should be balanced with the drag force to move the nucleus (Fdrag), as in the following force balance

[2]Fcentration + Fdrag=Fcfg.

The equation is converted into the following differential equation based on the linear relationship between the cellular force for the nuclear centration (Fcentration) and the distance from the center (x), as Fcentration = K × x (Fig. 3B), and between the drag force (Fdrag) and the velocity of the nucleus (dx/dt) as Fdrag = Cfric × (dx/dt) (Fig. 3C), as

[3]Cfric/K×(dx/dt) = -x + Fcfg/K.

Solving the differential equation of the force balance under centrifugation gives the following

[4]x(t) =(Fcfg /K) ×[1 – exp{-(t - t0)/τCK}],

where τCK = Cfric/K describes the ratio of Cfric to K, i.e., the relaxation time of the system, and t0 is the time at which the nucleus passes the cell center (i.e., x(t0) = 0). Eq. 4 is identical to Eq. 1, given Leq = Fcfg/K and τ = τCK. The good fit of the experimental data after the centrifugal speed reached its target values to the formula (Fig. 1D) supports the assumption that the force balance model with Fcentration = K × x and Fdrag = Cfric × (dx/dt) effectively describes the movement of pronuclei under centrifugation.

In the noncentrifugation experiment, similar to the force balance with centrifugation, the force balance without centrifugation (Fcentration = Fdrag) gives the following formula for the position of the nucleus without

[5]x(t) = x(t1) exp{-(t -t1)/τCK},

where t1 is an arbitrary time (e.g., t1 = 0). We fitted the trajectory of the nuclei (SI Appendix, Fig. S3) to the following

[5’]x(t) = x(t1) exp{-(t -t1)/τCK} + b.

Here, b is a fitting parameter describing the position where the nuclei settle after centration. Ideally, b should be 0 (i.e., the center), but because of the limitations in the accuracy of the measurement, we introduced b (Eq. 5′).

OI-DIC Microscopic Measurement and Data Analyses.

The OI-DIC microscope system developed at the Marine Biological Laboratory was used (25, 26). C. elegans embryos were cut out of adult worms in water, and embryos before the pseudocleavage stage were selected. Embryos were mounted on a coverslip (precoated with a 10% poly-L-lysine solution and dried). The coverslip was mounted on a glass slide with a spacer made of lanolin:paraffin = 1:1 (VALAP); therefore, the embryo was not compressed. Approximately 30 μL of water was added to fill the space between the coverslip and the glass slide, and the sample was sealed with VALAP. The sample was set on the OI-DIC microscope, equipped with a 40×/0.60 N.A. objective lens (LUCPlanFLN; Olympus) and a yellow 576 nm bandpass filter (FF01-576/10-25, Semrock, Rochester, NY). Images were recorded with a charge-coupled devices (CCD) camera (Teledyne Lumenera, Infinity3-1 M, Ottawa, ON, Canada) and processed to calculate the OPD as described (25, 26, 55, 56).

The obtained images with OPD values in each pixel were analyzed using ImageJ. A line with 10 pixels that passes through the center of a nucleus and is perpendicular to the long axis of the ellipsoidal embryo was drawn by hand. The OPD along the line was quantified using the “Plot Profile” function. OPD is defined as the difference in refractive index (ΔRI) multiplied by the thickness of the specimen (30). The OPD values along the line (x-axis) were considered to be the superposition of the following three

OPDbase = baseline,

OPDcyto = ΔRIC × 2 × {rC2 - (x -XC)2}0.5 (XC - rC ≦ x ≦ XC + rC),

OPDnuc = ΔRIN × 2 × {rN2 -(x - XN)2}0.5 (XN - rN ≦ x ≦ XN + rN).

Here, OPDbase, OPDcyto, and OPDnuc are the contributions to the OPD of the background (water), cytoplasm, and nucleus, respectively. ΔRIC and ΔRIN denote the differences in the refractive index of the cytoplasm and nucleus compared with that of water, respectively; rC and rN are the radii of the cytoplasm (cross-section) and nucleus, respectively; and XC and XN are the center coordinates along the x-axis of the cytoplasm and nucleus, respectively. The OPD profile was fitted to the superposition of the above functions using the Microsoft Excel solver tool. The obtained difference in refractive indices compared with those of water (ΔRIC and ΔRIN) was converted to the concentration of dry mass (Cdm, in g/mL) using Cdm = ΔRI/α, where α is 0.0018 (100 mL/g) (30). Finally, the density of the cytoplasm or nucleus (DC or DN, in kg/m^3^) can be calculated according to the following

D = (CW/100) × DW+10 ×Cdm,

where CW is the percentage of water in the cytoplasm or nucleus (% or g/100 mL), DW is the density of water (997 kg/m^3^ at 25 °C), and 10 × Cdm is the dry mass in kg/m^3^. CW can be determined as CW = (100 – Vsp × Cdm), assuming that the specific volume of the cytoplasm and the nucleoplasm is the same as the specific volume of proteins (Vsp = 0.75 mL/g) (30).

Confocal Microscopy Imaging to Measure the nuclear volume (NV) and Investigate the Distribution of Organelles.

To quantify the NV (SI Appendix, Fig. S2), images of embryos of strain XA3507 in which the nuclear membrane was fluorescently labeled were obtained. For the visualization of yolk granules after centrifugation (SI Appendix, Fig. S1), the embryos of strain RT130 were first mounted on the CPM, and after the centrifugation, the embryo was collected from the chamber and mounted on the confocal microscope as

Embryos were placed in 0.75× egg-salt buffer, and images were obtained at room temperature (22 to 24 °C) using a spinning-disk confocal system (CSU-X1; Yokogawa, Tokyo, Japan) mounted on an inverted microscope (IX71; Olympus). A 60×, 1.30 N.A. objective (UPLSAPO 60XS2; Olympus) was used for the experiments of SI Appendix, Fig. S2, whereas a 100×, 1.40 N.A. objective (UPLSAPO 100XO; Olympus) was used for the experiments of SI Appendix, Fig. S1. Images were acquired with a CCD camera (iXon; Andor Technology, Belfast, UK) controlled using Metamorph (version 7.7.10.0). Images were acquired with a z-interval of 1 μm. To quantify the NV (SI Appendix, Fig. S2), the outline of the pronuclei was traced by hand using Imaris (Oxford Instruments, Abingdon, UK). The surfaces of the pronuclei were reconstituted, and the volumes were calculated using Imaris.

Measurements of Nuclear Migration in Noncentrifuge Condition.

The speed of pronuclear migration in the noncentrifuge condition was quantified as Images of the wild-type N2 strain were obtained using a Nomarski DIC microscope (Olympus BX51) with a 100× oil immersion objective lens at a 2-s time interval. The two-dimensional coordinates of the center of the sperm-derived pronucleus were quantified manually using ImageJ.

Alternatively, we imaged a :mCherry; PH::GFP strain (CAL0332) with a confocal microscope as described above, with a 60× silicone immersion objective lens at a 5-s time interval with a 561-nm laser excitation. Before and after the time-lapse imaging of :mCherry, the cell membrane was visualized with a 488-nm laser excitation to check the cell boundary. Then, three-dimensional images were obtained, and the nuclei positions were quantified using Imaris software.

Because migration occurred mainly along the long axis of the embryo, the positions of the pronuclei were projected onto the long axis. The position of the center of the embryo was set to zero. The position(x)-vs.-time(t) plot (SI Appendix, Fig. S3) was fitted to Eq. 5′, using the Microsoft Excel solver tool. For fitting, we used the data from when the position exceeded 1/4th of the distance after the meeting point to the center of the embryo until the position settled near the cell center (SI Appendix, Fig. S3, filled circle).

Evaluation of Combined Errors.

The variance (error) of the calculated values was evaluated by the law of propagation of uncertainty. With this law, if the parameter y is a function of xi [i.e., y = f(x1, x2, …)], the variance of y (σ^2^(y)) is calculated as σ^2^(y) = Σi{(∂f/∂xi)^2^×σ^2^(xi)}. In this study, the variances of K and Cfric were calculated as see Table 1 for definitions of the

Fcfg = NV × Δρ × RCF×g,

K = Fcfg / Leq = NV×Δρ×g×RCF/Leq = NV×Δρ×g/βRL,

σ2(K) = (NV2 g2/βRL2)×σ2(Δρ) + (Δρ2 g2/βRL2)×σ2(NV) + (NV2Δρ2 g2/βRL4)×σ2(βRL),

Cfric = NV×Δρ×g/βRV,

σ2(Cfric) = (NV2 g2/βRV2)×σ2(Δρ) + (Δρ2 g2/βRV2)×σ2(NV) + (NV2Δρ2 g2/βRV4)×σ2(βRV).

Supplementary Material