Asymmetric Flows in the Intercellular Membrane during Cytokinesis

Abstract

Eukaryotic cells undergo shape changes during their division and growth. This involves flow of material both in the cell membrane and in the cytoskeletal layer beneath the membrane. Such flows result in redistribution of phospholipid at the cell surface and actomyosin in the cortex. Here we focus on the growth of the intercellular surface during cell division in a Caenorhabditis elegans embryo. The growth of this surface leads to the formation of a double-layer of separating membranes between the two daughter cells. The division plane typically has a circular periphery and the growth starts from the periphery as a membrane invagination, which grows radially inward like the shutter of a camera. The growth is typically not concentric, in the sense that the closing internal ring is located off-center. Cytoskeletal proteins anillin and septin have been found to be responsible for initiating and maintaining the asymmetry of ring closure but the role of possible asymmetry in the material flow into the growing membrane has not been investigated yet. Motivated by experimental evidence of such flow asymmetry, here we explore the patterns of internal ring closure in the growing membrane in response to asymmetric boundary fluxes. We highlight the importance of the flow asymmetry by showing that many of the asymmetric growth patterns observed experimentally can be reproduced by our model, which incorporates the viscous nature of the membrane and contractility of the associated cortex.

Introduction

Cell division in eukaryotes ends with the formation of two daughter cells. In eukaryotic cells, division is effected by a contractile ring that is positioned at approximately the equator of the mother cell through signaling between the anaphase spindle and the actomyosin cortex (1, 2, 3). The contractile ring then undergoes inward radial ingression driving membrane invagination and thereby cleavage furrow formation, which ultimately leads to constriction of the mother cell (4). In isolated cells such as Saccharomyces pombe, constriction proceeds until the formation of a narrow bridge between prospective daughter cells called the “midbody”, which upon abscission leads to complete separation of daughter cells. These cells divide via a pinch-off configuration, where the cell-cell contact area diminishes gradually (5). In contrast, in confined environments such as during embryo development (6, 7) and in epithelial tissues (8, 9, 10), upon cytokinesis, the daughter cells maintain a finite contact area and form a cell-cell interface consisting of two bilayer membranes, each belonging to one daughter cell (11, 12, 13). Formation of the contractile ring and the membrane partition require a supply of actin, myosin, and phospholipids. This material can either be directly recruited from the cytoplasm or can flow into the furrow through actomyosin-based cortical flows (14, 15, 16, 17). Cortical flows are known to facilitate alignment of actin filaments in the contractile ring (14, 18). Furthermore, their role in the context of spatially unrestrained cell division has been investigated previously (19). However, the role of cortical flows toward ring closure dynamics in confined environments is still unknown.

An interesting feature of cytokinesis in confined environments is that the cleavage furrow ingresses asymmetrically. How is the symmetry in the contractile ring broken? In C. elegans one-cell embryos and Drosophila pupal notum (8, 10), it was proposed that asymmetric ingression is a result of an asymmetric distribution of anillin and septin, which are major components of the contractile ring. Other mechanisms have been proposed in systems where a symmetric distribution of contractile ring components is observed. For example, in some epithelial tissues, anchorage of the ring in the apical side at the adherens junctions was shown to break the symmetry (11, 20). Dorn et al. (21), suggest that asymmetric ring closure can result from an enhancement of initial heterogeneities in the ring via positive feedback between membrane curvature and actin filament alignment. However, the role of cortical flows in facilitating asymmetric ingression is still unexplored.

Here, we develop a hydrodynamic model to investigate the combined effect of cortical flows and ring contractility on closure dynamics and asymmetric furrow ingression in confined cells. We show that an initially asymmetric ring can still close symmetrically if cortical flows are radially symmetric. Furthermore, we show that asymmetric ring closure can be achieved through radially asymmetric flows, and in cases where mirror symmetry of flows about all axes are broken, rotation of the contractile ring can be observed. Importantly, we support the findings of our model with experimental results from Caenorhabditis elegans one-cell embryos.

Materials and Methods

Model

As mentioned in the Introduction, lateral separation of the daughter cells is not possible in a spatially constrained environment caused by confinement (6, 12). This limitation causes the cell membrane of the mother cell to fold inward at the division plane to form the so-called membrane furrow (Fig. 1). The fold thus consists of two double-layer membranes weakly stapled together by adhesion proteins (3, 8, 9, 10). This membrane fold hangs inside the cell like an annular-shaped partition, with the ring at its inner edge. As the ring closes with time, this annular-shaped membrane goes on to form a full partition or the so-called “cell-cell interface”. In multicellular organisms, inside tissues, a stable cell-cell interface thus forms via binding of the two double-layer membranes by adhesion proteins like cadherin (8). This annular-shaped 2D interface grows as a result of material flow into it. The material flow is governed by the stress balance equation,

$$\nabla ·\sigma =0,$$ (1)

where the total stress tensor σ = σ_p + σ_a. The passive hydrodynamic component of the stress σ_p is given by

$${\sigma }_p=\eta \left[\nabla \mathbf{v}+{\left(\nabla \mathbf{v}\right)}^{\mathbf{T}}-\left(\nabla .\mathbf{v}\right)\mathbf{I}\right]+\lambda \left(\nabla .\mathbf{v}\right)\mathbf{I},$$ (2)

where η is the viscosity of the fluid, and σ_a is the active part resulting from the contractility of the underlying nascent acto-myosin cortex (22). We choose σ_a = σ₀I, an isotropic contractile stress, which is equivalent to a negative pressure. The shear and the bulk viscosities of the fluid are represented by η and λ, respectively—for 2D thin films λ ≈ 3η (23). We assume the contractile stress σ₀ to be uniform and therefore it does not affect the dynamics in the bulk of the film but appears in the boundary condition. At the inner boundary, where the contractile ring is present, the ring tension balances the stress in the film, as given by the following equation:

$$\mathit{\sigma }·\mathbf{n}=\kappa \left({\text{Σ}}_0-{\zeta }_L\kappa \mathbf{v}·\mathbf{n}\right)\mathbf{n}.$$ (3)

Here, Σ₀ and ζ_L are line tension and friction coefficient, respectively, of the contractile ring. The normal to the open inner boundary, directed toward the hole, is given by n, and κ is the curvature at the inner boundary, which evolves with time.

Figure 1.

Schematic diagram of the asymmetric septum closure based on (22, 28). (a) Given here is the longitudinal view and (b) x-y view of the shrinking hole located off-center. The r₀ is the initial furrow radius, and d is the distance between the initial center of the circular outline of furrow and center of the contractile ring.

We consider flow boundary conditions v(r₀) at the outer rim ρ = r₀. Cortical flows originating at the poles of the embryo and flowing toward the equatorial plane have been measured (14). These flows supply material to the growing cell-cell interface and appear as a planar inward flow at the outer periphery of the interface at ρ = r₀. The flow there is likely to have a dominant radial part v_ρ and may also have a nonzero azimuthal part v_θ to it. This inflow may also not be azimuthally symmetric. We consider various such possibilities (see Fig. 3) while solving Eqs. 1, 2, and 3. When the flow is radial and azimuthally symmetric, i.e., v(ρ,θ) = {v_ρ(ρ),v_θ = 0}, Eq. 1 reduces to η∂_ρ(∂_ρ + ρ⁻¹)v_ρ = 0 whose analytic solution was obtained in our earlier work (22). In our earlier work (22) we had explored the case that the growing interface receives phospholipid and actomyosin from the cytoplasm and not from cortical flow into the furrow. This was implemented by setting the boundary condition v(ρ = r₀) = 0, at the outer periphery. The resulting solution for the velocity field (obtained in (22)) showed that the divergence of interfacial flow field ∇.v was positive everywhere, indicating addition of material onto the interface. Inclusion of cortical flow, on the other hand, amounts to setting the boundary condition to v(ρ = r₀) = −v₀, with v₀ > 0. The solution for v_ρ(ρ) at ρ = r, the inner radius, yields the dimensionless closure speed:

$$\frac{dx}{d\tau }=\frac{\tilde{\sigma }x-1-\left(4\frac{\eta v_0}{{\text{Σ}}_0}\right)\frac{x}{1-x^2}}{\frac{1}{x}+\frac{2}{{\tilde{\zeta }}_L}\frac{1+x^2}{1-x^2}},$$ (4)

where x = r/r₀ and the dimensionless time is τ = t/τ₀, with τ₀ = ζ_L/Σ₀.The other dimensionless parameters in the equation are $\tilde{\sigma }$= σ₀r₀/Σ₀ and ${\tilde{\zeta }}_L$ = ζ_L/r₀η. Henceforth, to keep notations simple we will denote the inner radius by r and plot r/r₀ as a function of time. For late times x → 0, this solution yields the same exponential decay as in our earlier case with v(r₀) = 0. But for early times x → 1⁻, the initial closure speed is relatively higher due to the extra v₀-dependent term in the numerator. We have assumed v₀ to be constant over the whole course of closure. Typical cortical flow speed has been experimentally measured (14) to be about v₀ ∼ 0.1 μm/s and is typically not azimuthally symmetric. Using a similar value of v₀ in conjunction with other parameters, the v₀-dependent factor, 4ηv₀/Σ₀, is a small fraction and contributes significantly only when x → 1⁻, i.e., at very early times.

Figure 3.

Various initial and boundary conditions are categorized according to the symmetry of the incoming flow at the outer boundary and the initial location of the hole. The resulting flow structure is azimuthally symmetric for the left panel (a–d), and asymmetric for the right panel (e–i). However, all the figures in the right panel, except (i), have one mirror symmetry axis, shown by the dotted line. Rotation of the ring can be observed for (c) and (d) and (i). In (c) and (d), it is trivially due to the nonzero v_θ at the boundary, whereas in (i) it is due to lack of any mirror symmetry.

As a validation of our numerical code, we first verify that closure speed obtained from the numerical solution for this nonzero v₀ case matches with our analytic formula above (see Fig. 2), for an identical set of parameters. In (22), values of the above parameters were obtained by fitting the experimental data to the formula for closure speed obtained using v₀ = 0. However, we would like to point out that we made an error in (22) during our fitting of the experimental data. We made an assumption that $1/{\tilde{\zeta }}_L=0$ and determined σ₀ and τ₀ from the fit. Posteriori, we estimated that $1/{\tilde{\zeta }}_L=0.085$ and presented it as a justification for assuming $1/{\tilde{\zeta }}_L=0$. It turns out that numerically the resulting closure rates are quite different between these two cases. We now find that the same experimental data can be fitted well by minor variation of σ₀ and τ₀.

Figure 2.

The evolution of the hole radius and the corresponding velocity pattern for symmetric closure. (a) Here, we compare the analytic and numerical curves (both using the same parameters and v₀ = 0.084 μm/s) with the experimental data taken from (25) (same as that in (22)). The initial ring radius in the simulation was set at r = 13 μm, whereas the outer radius was at r₀ = 14 μm, same as in the experimental data. Therefore, the starting point of the simulation r/r₀ = 0.93 was matched with the experiment by shifting the simulation data by 20 s (see Numerical Solution and Methods for details). Here, (b) and (c) shows the evolution of the inhomogeneous velocity pattern, where (b) is for intermediate time when the closure speed—slope of the curves in (a)—is high and (c) is for the terminal slow phase.

We find that inclusion of experimentally relevant values of v₀ in Eq. 4, and independently estimated new parameters, give a reasonable fit to the experimental data on symmetric closure (see Fig. 2). Therefore, a good data fitting to the closure speed in the symmetric case cannot establish the importance of the cortical inflow v₀. Instead, we explore the possible role of v₀ in asymmetric closure. Sticking to these same parameters, we numerically investigate how azimuthal asymmetry in v₀ may lead to new qualitative features in the asymmetric positioning of the hole, because no analytical expressions are available in this case.

For our general solution for v(ρ,θ), all the components of the 2D viscous stress tensor σ are nonzero and dependent on both ρ and θ. We solve for v numerically, as described below, using Cartesian x,y coordinates.

Numerical solution and methods

The numerical solution to the governing partial differential equation (Eq. 1) along with the boundary conditions is obtained using the finite element analysis (FEA). The FEA is implemented using the commercial finite element software Comsol Multiphysics (https://www.comsol.com; refer to the Supporting Material for details) with an Arbitrary Lagrange Euler scheme to account for the moving internal boundary.

Model definition and parameters. We input the governing Eq. 1 using the standard coefficient form in the Comsol basic module. The basic geometrical parameters are gleaned from the experimental data (24). The initial ring radius is taken as r = 13 μm, which decreases as the ring contracts, whereas the initial furrow remains at a constant value of radius r₀ = 14 μm. For our numerical explorations in this article, we have independently estimated the parameters. In particular, the values for active line tension (Σ₀), active surface tension (σ₀), effective acto-myosin viscosity (η), and contractile ring friction coefficient (ζ_L), are Σ₀ = 3.3 × 10⁻⁹ N, σ₀ = 2.91 × 10⁻⁴ N/m, η = 0.5 × 10⁻³ N.s/m, and ζ_L ≃ 0.825 × 10⁻⁷ N.s, respectively. In Fig. 2 a, we shift the simulation data to the right, keeping r/r₀ ≈ 1 to prevent possible confusion when compared with the same experiment data (25), which is used in (23) as well. For all other simulation data, unique to this article, we consistently start with r(0)/r₀ ≈ 0.93, corresponding to r(0) = 13 μm—in a sense, that is how we define t = 0. For quantitative comparison of our numerical results to experimental data on asymmetric closure, we vary v₀, Σ₀, σ₀, and ζ_L (see Supporting Material).

Boundary conditions. Dirichlet boundary condition with varying external velocity, v₀, is applied at the outer boundary. The boundary condition on the contractile inner ring Eq. 3 is supplied with the flux source boundary condition for momentum option in the software Comsol. As a part of the solution, the inner boundary is moved along the normal direction (n) with a speed of v_in·n, where v_in is the flow velocity at the contractile ring.

Mesh generation and numerical solution. For better convergence to the solution, a triangular mesh is chosen with appropriate refinement and the element size is calibrated for fluid dynamics. The geometric shape order is set to quadratic and Laplacian smoothing technique applied for avoiding inverted mesh elements. We developed a strategy to prevent mesh distortion and optimize the simulation time by using Comsol Livelink for MATLAB (The MathWorks, Natick, MA). The model is simulated for a time duration of Δt = 0.1 s in Comsol and then paused. To ensure good mesh quality, a circle is fitted to the inner moving boundary points and a new geometry for the septum is freshly recreated. Fitting of a circle to the inner set of points is for simplicity and is supported by experimental evidence presented in Fig. 5 a. This process is repeated using an automated MATLAB script until the radius of the inner ring becomes negligible compared to the outer radius of the septum. This procedure of frequent fitting and remeshing ensures high mesh quality and ensured numerical accuracy of our simulations. We used the analytical solution for the radially symmetric problem described earlier (see Fig. 2) to validate our present Comsol FEA scheme, which was designed and used to tackle problems with different initial and boundary conditions.

Figure 5.

Comparison of the numerical data with experimental results. (a) Fluorescence images of asymmetric septum closure in C. elegans embryo on the septum (see Movie S1). (b) Numerical solution of the Eq. 4 showing time course of asymmetric septum closure with asymmetric radially inward velocity: v₀ = 0.2 μm/s on the lower half, and v₀ = 0.02 μm/s on the upper. The initial hole was also displaced off-center along the positive y axis. As a result of the flow the center of the hole moves further up in the beginning and later the hole closes due to strong line tension in the ring (see Movie S2). (c) The curve shows the time evolution of the radius of the contractile ring. The circle and the line indicates the experimental data and numerical results, respectively. (d) The graph shows the extent of asymmetry in ring closure with time, where d is the distance between the initial center of the circular outline of the furrow and the instantaneous center of the contractile ring and r₀ = 14 μm is the initial furrow radius (see Movie S1).

C. elegans strains. In this study, the LP133 strain (nmy-2(cp8[NMY-2::GFP + unc-119(+)]) I; unc-119(ed3) III) (26) was used for imaging cytokinesis. C. elegans worms were cultured on OP50-seeded NGM agar plates as described (27).

Mounting and image acquisition. Worms were dissected in M9 buffer and the embryos were mounted on 0.5% agarose on a coverslip. An eye-lash tool was then used to rotate the embryo to obtain an end-on view i.e., viewed from either the anterior or posterior pole. Time-lapse movies of cytokinesis were acquired at 23–24°C with a spinning disk confocal microscope using a C-Apochromat 63×/1.2 NA objective lens (Carl Zeiss, Oberkochen, Germany), a model No. CSU-X1 scan head (Yokogawa Electric, Tokyo, Japan), a 525/50$\text{nm}$ bandpass emission filter (Semrock, Rochester, NY) and a Neo sCMOS camera (2560 × 2160 pixels; Andor Technology, Windsor, CT). A stack consisting of 10 z planes (1 μm apart) with a 488$\text{nm}$ laser (50% AOTF intensity) and an exposure of 100 $\text{ms}$ was acquired at an interval of 3 $\text{s}$ from the onset of cytokinesis until ring closure. The maximum intensity projection of the stack at each time point was then subjected for further analysis.

Results

The analytical result obtained in Fig. 2 pertained to the simplest case of ring closure when the ring closure is radially symmetric, and the cortical flows are exclusively radial. As we mentioned before, in the case of confined cell division, for example, in C. elegans, it has been observed that the hole in the closing partition is typically located off-center and remains so as the closure proceeds. Flows on the partition in such cases are expected to have both azimuthal and radial components. Nonsymmetric ring closure has been attributed to the azimuthal asymmetry in anillin and septin distribution in the contractile ring (28). Removal or downregulation of anillin results in symmetry being restored, although the actual molecular mechanism of how anillin and septin connections between the membrane and the ring holds part of the ring hinged to the periphery whereas the rest of the ring contracts is unclear. Dorn et al. (21) suggest that the asymmetry in the ring closure is caused due to the enhancement of initial heterogeneities in the ring closure by positive feedback between the membrane curvature and actin alignment at the cytokinetic furrow. They further observe that interplay among filament bundling, curvature-dependent filament alignment, rate of filament alignment, and speed of filament sliding can modulate the ring closure pattern between symmetric and asymmetric. This model, although extremely detailed, does not incorporate either the actomyosin flows that are observed outside the furrow, or the presence of cell-cell partition between the daughter cells. We, on the other hand, focus on the asymmetry of the growing interface resulting from the initial eccentricity of the inner ring and the azimuthal heterogeneity of the cortical influx to the partition. We wish to understand not only the closure times for the ring as described in Fig. 2, but also to study various ring closure and cortical flow patterns as are experimentally known to be present (see Fig. 5 a). To this end, we explore different instances of boundary influx conditions and their effect on ring closure—the different scenarios explored below are presented in Fig. 3.

Asymmetric ring closure

We next probed the dynamics of ring closure in an initially off-centered ring in the presence and absence of cortical flows. We first discuss the case presented in Fig. 3 e, where the system with an off-center ring at initial time is supplied with uniform external velocity v₀ at the outer boundary (see Fig. 3, e and f). We perform numerical simulation for both zero (v_ρ(r₀) = 0) and nonzero (v_ρ(r₀) = 0.084 μm/s) values of radial boundary velocity for the flow. It can be observed that the ring remains off-center until it contracts wholly. Fig. 4 shows different positions of the ring when it was initially asymmetric and v_ρ(r₀) = 0. The important finding of this exercise is that, although the contractile ring demonstrates off-center closure, the center of the ring remains stationary during the closure.

Figure 4.

Off-center hole remains off-center for v₀ = 0. A uniform radial inflow is imposed throughout the outer boundary of an off-center hole. Here, (a) shows the initial off-center hole and (d) is the final position of the closed hole, which is still off-center. Here, (b) and (c) are the velocity plots at intermediate times during the ring contraction. The resulting flow in the interface has only a radial (v_ρ) component. The hole moves off-center and closes itself off-center under the strong action of the contractile ring. Because of one mirror symmetry, the inner boundary does not have any net rotation.

Asymmetric ring closure with migration away from the center

We next asked whether an asymmetry in cortical flows can result in asymmetric ring closure accompanied by ring translation. To provide a simple explanation for a particular instance of the ring closure pattern observed experimentally in Fig. 5 a, we first explore the case depicted in Fig. 3 g, in which the contractile ring is located centrically but the flows at the boundary are asymmetric. Specifically, we have provided an external velocity v₀ = 0.2 μm/s on the lower-half, and v₀ = 0.02 μm/s on the upper-half of the outer boundary of the septum (see Fig. 5 b). The time taken by the ring to contract in our model matches reasonably well with the experimental data for an asymmetric closure with radially asymmetric inward velocity (Fig. 5, a and b). This is further quantified in Fig. 5 c, which shows the time evolution of the radius of the contractile ring, indicating the experimental data and numerical results (circles and line, respectively).

Initially with a centrally placed ring, the contraction proceeds with a displacement of the ring toward the boundary (toward the positive y axis in Fig. 5 a). The center of the ring steadily moves up until the effect of ring contractility becomes strong enough to speedily close the gap. Fig. 5 d shows the extent of asymmetry during the contraction of the ring with increasing time. The asymmetry parameter is quantified with d/r₀ (see Fig. 1), where r₀ = 14 μm is the initial furrow radius, and d is the distance between the initial center of the circular outline of the furrow and the instantaneous center of the contractile ring (21). This asymmetry parameter is an indicator of the relative off-center movement of the contractile ring from its initial central position. As can be seen from Fig. 5 d, the experimental data shows nonmonotonicity in the asymmetry parameter, which is not reproduced in our simulation—possibly due to a variety of reasons. One reason, for example, may be that the ring contractility and flows may be time dependent. It is also likely (see Fig. 5 a) that in the experiment, part of the ring transiently gets anchored to the boundary whereas the rest of the ring contracts, making the ring-center drift toward the boundary. This could be due to asymmetric anchoring of the ring by anillin. Our simple model does not incorporate any of these possible causes for the nonmonotonic ring migration behavior. In our simulation, the ring migrates marginally only in the initial stage and then stays put. However, even in the experiment, at the late stage, the ring gets released from the boundary and migrates closer to the center. This effect brings down the asymmetricity parameter to a value that is close to the saturated value of the asymmetricity parameter obtained in our simulation. As Fig. 5 b shows, the basic phenomena of asymmetric ring closure is indeed captured by our model, albeit qualitatively.

Rotation of contractile ring during closure

Interestingly, we observed that the contractile ring undergoes subtle rotations during closure (see Movie S1). We wondered whether an asymmetry in flows can lead to the observed rotations. Such flow asymmetry can arise in our model as a result of either the presence of tangential component of the flow at the outer boundary of the furrow (Fig. 3, c and d), or from the lack of both azimuthal symmetry and mirror symmetry of the boundary flux with respect to the initial position of the contractile ring (Fig. 3 (i); Movie S3 and Fig. S1). We now explore the effect of these boundary conditions on the rotational dynamics of the contractile ring.

Time-independent boundary conditions are set by fixing the velocity profile v = (v_ρ,v_θ) and their θ-dependence, at the outer periphery (ρ = r₀). Position of the hole at time t = 0 sets the initial condition. The azimuthal symmetry of the resulting flow can be broken either by choosing the hole to start off-center (the center being defined as the center of the outer circle), or by making the v(r₀) dependent on θ. An off-centered hole can be achieved in two ways—by moving it either (a) along the x or y axis, which still preserves one mirror symmetry axis or (b) along both x and y axes, which spoils all mirror symmetries. A combination of boundary and initial conditions can also result in loss of all mirror symmetries resulting in a θ-dependent flows. For example, first displacing the initial hole along y axis (keeping mirror symmetry about y) and then choosing the boundary condition to be different on left and right halves, destroys the y symmetry (see Fig. 3 (i)). Only in the case where mirror symmetry about both x or y axes is broken, does the hole start to rotate. Therefore, the configuration in Fig. 3 (i) generates rotation, whereas the one in Fig. 3 (h) does not. When a nonzero v₀ is applied on one-half of the outer periphery with an off-center ring, the ring rotates along with a drift in the position due to absence of mirror symmetry (Fig. 6, a–c). Such rotations can, of course, be generated by choosing nonzero v_θ at the boundary (see Fig. 3 d), but this case does not lead to eccentricity in the ring position. Fig. 6 d shows a central hole with nonzero radial and azimuthal v₀ components supplied, experiencing rotation of the ring. Interestingly, this simple case can mimic the overall embryo rotation that is observed for a window of a few tens of seconds in C. elegans (29, 30). We thus provide various scenarios that can cause rotation of the contractile ring, and demonstrate their plausibility with numerical simulations.

Figure 6.

Rotation of the central hole. Here, (a–c) show evolution of a maximally asymmetric flow where all mirror symmetries are violated by (i) choosing a nonzero radial inflow on the left half, and (ii) displacing the initial hole along the y axis. Nonzero v_θ is generated in the interior, although the input at the boundary has v_θ = 0. Absence of any mirror symmetry leads to net rotation of the hole in the counterclockwise direction. We computed the sum of v_θ along the inner boundary to check this quantitatively (see Fig. S1). (d) Shown here is a display of rotation that is trivially generated by choosing uniform nonzero v_θ at the outer boundary ((c and d) in Fig. 3).

Ring closure dynamics for all conditions

Fig. 7 summarizes results from all our simulations and compares them with experimental data on the ring closure rate. As can be seen, the simulation curves are quite close to each other irrespective of different initial and boundary conditions listed in Fig. 3, and also with respect to variations in some of the parameters used for Figs. 2, 4, 5, and 6. The solid circles in this figure represent experimental data collected by averaging over six independent experimental runs. The deviation band shown in the figure result from the experimental data. This figure indicates robustness of the closure rate irrespective of various boundary and initial conditions.

Figure 7.

The closure rate is robust with respect to various initial and boundary conditions listed in Fig. 3. Data for radius versus time, from all our simulations (open symbols), are compared with our experimental data (solid circle). The various initial and boundary conditions for the simulation runs are listed next to the respective open symbols. The deviation band results from averaging the experimental data over six independent movies. The band ends where the hole size becomes too small for a reliable size estimation.

Discussion

In this article, we developed a hydrodynamic model to investigate cytokinetic ring closure in C. elegans. Using a simple description applicable just for the interface between the daughter cells, we obtain a variety of features observed in the closure plane. The major contention of our article is that various asymmetries in cortical actomyosin flows from the poles toward the ingressing furrow can lead to asymmetric closure as well as different movement patterns of the contractile ring. For example, we demonstrate that asymmetric actomyosin influx between the two halves of the furrow plane can lead to an overall migration of contractile ring away from the center of the division plane. A further break in the symmetry of the boundary influx along the azimuthal direction on the division plane can lead to the rotation of the contractile ring. Along with the ring closure times, our model is able to recapitulate a variety of experimentally observed ring closure patterns in the division plane of C. elegans.

It is known that an asymmetric distribution of contractile ring components such as anillin and septin facilitates asymmetric closure (10, 28). Furthermore, asymmetries in anillin distribution is likely to result in polarized recruitment of cortical components such as myosin (31, 32, 33). Given that gradients in cortical contractility drive spatially heterogeneous flows in C. elegans (34), asymmetry in distribution of cortical components can possibly influence azimuthally asymmetric cortical flows, thus leading to asymmetric ring closure. Furthermore, a very recent article by Reymann et al. (14) demonstrated that the anillin deficiency in C. elegans leads to a significant slowing of cortical flows. As per our findings, the strength of asymmetry of cortical flows at the outer boundary remains the main driving force for asymmetric ring closure. When the cortical flows slow down, their overall effect on partition closure diminishes. Ring contractility, which would then be the main force for partition closure, is not sufficient for inducing asymmetry. We therefore provide a plausible reason as to why anillin deficiency may lead to symmetric ring closure.

There is another nontrivial effect that may arise due to the cortical influx v₀. Basic physical considerations indicate that increase in fluid viscosity η should slow down the ring closure dynamics. However, as can be seen from Eq. 4, the presence of cortical flows can actually increase the rate of ring closure due to the presence of term v₀η in the numerator. The counterintuitive influence of η on ring closure can be further enhanced during asymmetric ring closure, and indeed, increase in η, while keeping other parameters to be the same, leads to relatively faster closure of the ring and builds up greater asymmetry during ring closure (see Fig. S4). However, this may be difficult to realize experimentally because, although η can be tuned by varying temperature, say, it may be difficult to simultaneously maintain constancy of the other parameters. Nevertheless, we mention this finding to emphasize the counterintuitive influence of boundary influx on the ring closure dynamics via fluid viscosity. It is also noted that the membrane thickening will be spatially heterogeneous due to nonuniformity of the velocity field. However, for the purposes of simplicity, we do not explicitly consider it in our current model (see Supporting Material for details).

The mechanisms of ring closure asymmetry discussed here are different than those proposed by Dorn et al. (21), and so is our overall approach. In their model, the timescale of sliding and alignment of actin filaments at the furrow with respect to membrane curvature was proposed to be important in facilitating asymmetric ring closure. However, their model does not explicitly account for the presence of an annular partition with finite contact area between the two daughter cells. Therefore, it is not quite clear how the Gaussian curvature at the furrow ingression can be explicitly linked to the dynamics of the ring, which is present at the inner edge of the partition (see Fig. 1). Moreover, it was recently demonstrated by Reymann et al. (14) that the compression of cortical flows, resulting from the spatial flow gradient, is sufficient to cause alignment of actin filaments, and is crucial for the cytokinesis process (the role of such cortical flows is not accounted for by Dorn et al. (21)).

In our model, we do not explicitly describe the process of actomyosin alignment leading to the initiation of the contractile ring, which we model as a line with a given contractility and internal friction. Instead, we focus on modeling of the effect of boundary flows at the furrow partition on the overall ring closure and movement patterns. Similarly, we do not explicitly include anillin in our model, but the off-center location of the ring, which appears as an initial condition in our simulation, partly serves as a proxy for the factors that cause asymmetry. Moreover, it is worth mentioning that our model is capable of producing the same closure time (7) for smaller cell sizes provided the line tension and the internal friction of the ring linearly scale with the cell size (22). In conclusion, we have shown that the asymmetry in the cortical influx into the septum can be one of the important causes behind asymmetric septum closure; however, it may not be a necessary condition.

Author Contributions

A.S. and M.M.I. designed and supervised research. V.V.M., A.A., and S.S.S. developed mathematical and finite element framework. V.V.M. performed all numerical simulations. S.R.N. performed experiments. M.M.I., A.S., S.R.N., and V.V.M. wrote the manuscript.

Editor: James Grotberg.

Supporting Citations

References (35, 36, 37, 38, 39) appear in the Supporting Material.

Supporting Material

Movie S1. Cytokinetic Ring Closes in an Asymmetric Fashion

A dual-colored transgenic line (NMY-2::GFP; PH::mCherry), generated by crossing LP133 strain with PH::mCherry (from the Hyman lab, MPI-CBG, Dresden, Germany), was used to image cytokinetic ring closure. The embryo was oriented in an end-on view (from anterior or posterior pole) and multiple slices (1 μm apart) were obtained. The maximum intensity projection is displayed here. Note that the ring predominantly closes in an asymmetric fashion. (Left) NMY-2::GFP marks nonmuscle myosin 2; (right) PH::mCherry marks membrane. The total closure time is 250 s for this embryo.

Movie S2. Migration of the Hole under Asymmetric Flow

This movie was generated from our simulation using the parameters and the initial and boundary conditions described below. Values for the internal ring tension (σ₀), effective acto-myosin viscosity (η), line tension (Σ₀), and contractile ring friction coefficient (ζ_L) were chosen to be σ₀ = 2.5 × 10⁻⁴ N/m, η = 10⁻³ N.s/m, Σ₀ = 2.8 × 10⁻⁹ N, and ζ_L = 0.495 × 10⁻⁷ N.s. Initial radius of the contractile ring was set to be r = 13 μm and the outer boundary was set at r₀ = 14 μm. The outer boundary was fed with a radially inward flow, with velocity v₀ = 0.2 μm/s on the lower-half, and v₀ = 0.02 μm/s on the upper-half. It can be observed that the initial hole was displaced off-center, along the positive y axis, as time progressed. The blue arrows denote the associated cortical flows. As a result of the flow, the center of the hole slowly migrated upward at early times and later predominantly underwent closure due to strong line tension effect. The total time duration of this numerical experiment until the hole is closed to around 99% is ≈170 s.

Movie S3. Rotation of the Ring under Asymmetric Flow that Broke All Symmetries

This movie was generated in the simulation to understand the ring closure dynamics for the system configuration presented in Fig. 3(i). The various parametric values are σ₀ = 3 × 10⁻⁴ N/m, η = 10⁻³ N.s/m, Σ₀ = 4.4 × 10⁻⁹ N, and ζ_L = 1.65 × 10⁻⁷ N.s. The initial radius of the contractile ring was r = 10.5 μm and it was placed off-center. The outward boundary was provided with a radially inward flow, with velocity v₀ = 0.5 μm/s on the lower-half, and v₀ = 0 .05 μm/s on the upper-half. This particular case was designed to break both azimuthal symmetry and all possible mirror symmetries by appropriate choice of boundary flux and initial position of the ring, and consequently it led to rotation of the contractile ring. Because our simulation involved repeated remeshing (see Materials and Methods), the points on the inner boundary were not individually tracked, as a result of which the rotation is not apparent in the movie. However, we computed the line integral of the tangential speed on the inner boundary (ring) and it produced a nonzero value signifying rotation. The total closure time was ≈305 s.