Stability of asymmetric cell division: A deformable cell model of cytokinesis applied to C. elegans

Abstract

Cell division during early embryogenesis is linked to key morphogenic events such as embryo symmetry breaking and tissue patterning. It is thought that the physical surrounding of cells together with cell intrinsic cues act as a mechanical “mold,” guiding cell division to ensure these events are robust. To quantify how cell division is affected by the mechanical and geometrical environment, we present a novel computational mechanical model of cytokinesis, the final phase of cell division. Simulations with the model reproduced experimentally observed furrow dynamics and describe the volume ratio of daughter cells in asymmetric cell divisions, based on the position and orientation of the mitotic spindle. For dividing cells in geometrically confined environments, we show how the orientation of confinement relative to the division axis modulates the volume ratio in asymmetric cell division. Further, we quantified how cortex viscosity and surface tension determine the shape of a dividing cell and govern bubble-instabilities in asymmetric cell division. Finally, we simulated the formation of the three body axes via sequential (a)symmetric divisions up until the six-cell stage of early C. elegans development, which proceeds within the confines of an eggshell. We demonstrate how model input parameters spindle position and orientation provide sufficient information to reliably predict the volume ratio of daughter cells during the cleavage phase of development. However, for egg geometries perturbed by compression, the model predicts that a change in confinement alone is insufficient to explain experimentally observed differences in cell volume. This points to an effect of the compression on the spindle positioning mechanism. Additionally, the model predicts that confinement stabilizes asymmetric cell divisions against bubble-instabilities.

Significance

A crucial morphogenic step during early embryonic development is symmetry breaking in the embryo. For C. elegans the formation of the three body axes can be traced back to the six-cell stage, where tissue topology is the result of symmetric and asymmetric divisions. How cell mechanical boundary conditions and cell intrinsic cues influence this process of symmetry breaking is still an open question, as currently, a quantitative mechanical description of cytokinesis in complex architectures is lacking. We present a simple mechanical model of cell division, incorporated in an existing mechanical cortex model and simulated cytokinesis in geometrically confined environments. Our approach was able to both capture furrow ring dynamics and predict the volume ratio of daughter cells accurately. By simulating early C. elegans development with different geometrical boundary conditions, we were able to trace back the origin of volume discrepancies between the experimental setups to a quantifiable shift in spindle positioning during cytokinesis. Finally, we showed how embryo confinement partially stabilizes bubble-instabilities that arise during asymmetric cell division during the early cleavage phase.

Introduction

Early embryogenesis is characterized by a well-orchestrated sequence of cell division and differentiation events. In bilateria, an important step during this process is the establishment of the three main body axes: anterior-posterior, dorso-ventral, and left-right (1). In model organisms with an invariant developmental cell fate, such as the nematode Caenorhabditis elegans, the origin of these symmetry-breaking steps has been traced back to asymmetric cell divisions, which can be biochemical and/or physical in nature (2,3). Although the importance of physical asymmetry, such as an asymmetric volume ratio of daughter cells, remains poorly understood, recent observations for P0 divisions in C. elegans have linked daughter cell size asymmetry to changes in temporal cell cycle coordination, tissue topology, cell fate, and even overall robustness of early embryogenesis (3).

To control the volume of daughter cells, cell division relies on the precise positioning of the spindle apparatus, as the future cleavage site aligns with the spindle midplane (4,5,6). Accumulated discrepancies during this process can lead to polyploidic cells, which have major detrimental effects on tissue homeostasis and development (7,8,9). It is thus imperative for a dividing cell to have strict spatiotemporal control of its spindle, as any shift in spindle position influences the volume ratio daughter cells and viability (10). Multiple spindle positioning mechanisms have been identified and categorized based on the nature of the underlying cues, e.g., internal versus external and biochemical versus mechanical (11). Spindle positioning in the P0 division of C. elegans for example, which is a highly asymmetric both in volume and distribution of molecular species, is governed by a biochemical gradient that induces a bias in the magnitude of cortex-spindle force interactions toward the posterior pole (12). For wild-type C. elegans, this bias produces a net spindle division plane offset, from the cellular center of mass in the direction of the posterior pole. This offset eventually results in an asymmetric division where the anterior cell is significantly larger than the posterior cell, at a $\pm 60/40$ volume ratio. Disrupting biochemical cues, e.g., the distribution of PAR proteins, or interfering with mechanical cortex-spindle interactions can lead to symmetric P0 division (3,13). Interestingly, C. elegans embryogenesis seems robust to compression-induced eggshell deformation (14). However, in these geometrically perturbed setups, changes in cellular movement (14) and enhanced volume asymmetry in divisions that contribute to the LR-axis (15) have been observed in the early embryo.

Mechanically, the final phase of cell division, cytokinesis, is driven by a tensile furrow ring consisting of cross-linked actin filaments (16). The dynamic cross-linking of tread-milling actin subunits in the ring is thought to drive tension buildup, resulting in furrow ingression (17). As of yet, although key biochemical mechanisms that regulate cytokinesis have been identified (2,18,19,20,21), a full spatiotemporal mechanical description of cytokinesis at cell scale is still lacking. The past decade has, however, seen significant progress in the development of mathematical and computational models that describe the physics of cytokinesis. However, most fail to account for the geometrical and mechanical constraints imposed by the surrounding tissue architecture and the embryonic boundary. Furthermore, few mechanical models exist that are able to represent the physics of asymmetric cell division (22,23). Yet, especially in early embryonic development, it is thought that the coupling between the asymmetry of cell divisions and the geometry and connectivity of the surrounding tissue stabilizes robust tissue patterning (24,25,26,27,28).

At long timescales, the model of a cell as a fluid droplet with surface tension and adhesive bonds provides a minimal description of the mechanical response of a cell. It can be tuned with a small set of experimentally accessible parameters, such as the interfacial tension and the effective cortical tension (29,30). However, droplets under tension are inherently susceptible to “bubble-instabilities,” where regions with locally elevated curvature, such as the small daughter half in an asymmetric division, tend to collapse in favor of low curvature regions due to differences in local pressure. Interestingly, such instabilities have been observed in vitro in HeLa cells and fibroblasts (31), where they produce shape oscillations during cytokinesis.

We present a novel computational model of cytokinesis based on an existing deformable cell model (DCM) (32,33,34,35). In the DCM, the cell cortex is approximated as a thin, viscous shell under surface tension and cell-cell interactions are taken into account through adhesion forces. For this work, we made abstraction of the underlying molecular mechanisms of furrow ring mechanics and used a coarse-grained representation where the furrow ring was approximated by a ring that shrinks as cytokinesis progresses (36) and thereby enforces cytokinesis to occur; see model section. Furrow ingression then follows as a result of tension buildup in the contractile ring and local deformation of viscous cortical material. Ring contraction progresses until cell abscission is initiated at the prescribed abscission radius. After cell abscission, membrane and cortex wound healing wrap up cell division, splitting the mother cell into daughter cells (37). C. elegans was used as our biological model system due to its invariant developmental path and transparent eggshell, allowing for reliable and direct volume estimates for each cell. Furthermore, significant effort has been made in mechanically characterizing C. elegans, permitting proper model calibration and relevant quantitative comparisons between simulation results and experimental measures.

Materials and methods

Cell mechanical model

In line with previous models (38,39,40,41), our mechanical model considers tissues as foam-like materials. In this minimal description, a cell is modeled as a volume-conserving, pressurized bubble, characterized by a thin, viscous shell that is under an active tension (42). This shell represents the combined acto-myosin cortex-membrane complex of the cell, further denoted as “cortex,” which is assumed to dominate the mechanical response in deformation and in cell-cell interactions, and is parameterized by thickness t, surface tension γ, and viscosity η. Cells adhere with adhesion energy ω, binding them to each other and determining the contact angle at the cell-cell interface; see Fig. 1 A. Furthermore, cell-cell and cell-substrate interactions include a wet friction with friction coefficient ξ. We use a particle-based computational model to simulate the mechanical behavior of a collection of cells. To this end, we represent the shell that represents the cortex-membrane complex using a triangulated surface mesh where nodal positions act as the relevant degrees of freedom; see Fig. 1 A. Cell volume is conserved using an apparent bulk modulus $K^{\prime }$ that gives rise to an internal pressure (32,33,34). A detailed description of the mechanical model can be found in the Supporting material. We verified that the model reproduces contact angles as predicted by the Young-Dupré equation, cell-cell pull-off forces as predicted by the Brochard-Wyart and de Gennes derivation (43), the shear strain rate in function of applied stress, and the viscous relaxation dynamics upon pressurization (Supporting material p. 7–10).

Figure 1.

Cortex and furrow ring model. (A) Viscous cortex model based on foam-theory. (B) Furrow ring model, approximated as closed loop consisting of shrinking (depolymerizing) elastic subunits. As the subunits shrink, an elastic force is generated, leading to a net inward facing force that constricts the cell cortex. (C) Schematic representation of parameters that define asymmetric cell division in a confined environment. To see this figure in color, go online.

Phenomenological model of the furrow ring

Carvalho et al. observed furrow ingression rates to be constant, until the ring reaches a threshold radius $R_t=3-3.6\mu \mathrm{m}$. Beyond this threshold, the furrow ring’s interactions with interpolar microtubules become nonnegligible, slowing down the rate of ingression before reaching a halt at radius $R_f=0.5-1.0\mu \mathrm{m}$(36). Based on these observations, we modeled the furrow ring as a contracting entity consisting of subunits that decreases in resting length at a constant velocity (17,44,45); see Fig. 1 B. Since furrow-microtubule interactions are not modeled, an effective subunit depolymerization velocity was introduced as

$$\frac{\mathrm{d}{l}_{s,0}}{\mathrm{d}t}=\{\begin{array}{cc}-v_d\text{,}& l_s\ge \frac{l_0{R}_t}{R_0}\\ -v_d\frac{l_{s,0}{R}_0}{l_0{R}_t}\text{,}& \frac{l_0{R}_t}{R_0}>l_s>\frac{l_0{R}_f}{R_0}\\ 0& l_s\le \frac{l_0{R}_f}{R_0}\text{,}\end{array}$$ (1)

with $v_d=\frac{l_0}{{\tau }_d}$, initial subunit (resting) lengths $l_0=\pm 600\text{nm}$, instantaneous subunit length $l_s$ and resting length $l_{s,0}$ and total depolarization time ${\tau }_d$ (17,36). By attributing a mechanical stiffness to the ring, the imposed contraction generates a line tension and thereby drives the shape change of the viscous DCM. From Eq. 1, the contractile line tension $F^f$ generated in the furrow ring due to depolymerization of the subunits can be written as

$$F^f=k\left[\frac{l_s}{l_0}-\left(1-\frac{t}{{\tau }_d}\right)\right]\text{,}$$

assuming a linear-elastic model with relative stiffness k (see Supporting material p. 12–13). Substituting $l_s=2\pi r_f/N$ and $l_0=2\pi R_0/N$, with $r_f$ the instantaneous furrow radius, it can be seen that $F^f$ is independent on initial cell size. As tension builds up in the furrow ring, the cortex deforms, resulting in furrow ingression. Since in this work we assume a sufficiently stiff contractile ring, i.e., high k, the furrow shape is effectively prescribed and will follow Eq. 1; see further Figs. S7–S9 in the Supporting material. Cell abscission is triggered when the furrow perimeter reaches a set threshold value, $2\pi R_f$. During this phase, furrow ring components are disassembled, and the mother cell is split into two new entities, where the daughter cells inherit the mechanical properties of the mother cell.

Model of asymmetric cell division

During cytokinesis, spindle positioning determines the volume ratio of future daughter cells (6). We aim to simulate the effect of spindle position on the dynamics of cytokinesis, without modeling in detail the underlying physical mechanisms that drive the position and orientation of the mitotic spindle (46,47). Hence, we propose a phenomenological description of the division plane positioning mechanism that imposes the spindle offset α and orientation angle φ relative to, respectively, the cell’s center of mass and the division plane; see Fig. 1 C.

Asymmetric division of a viscous tensile shell is inherently mechanically unstable due to polar contractility, resulting in the bubble-instability akin to the two-balloon experiment (31,48). Physically, this instability can be evaded if the cortex is sufficiently viscous such that the cell is able to complete cytokinesis before the instability or if mechanical structures are present that resist a complete daughter-half collapse. For the latter, microtubule-cortex interactions provide a possible candidate, as microtubules can provide both mechanical rigidity and active force generation (49). In line with the work of Sedzinski et al. (31), our model does not explicitly introduce these interactions, but it introduces a correction pressure $P_D^{\mathit{div}}$ that prevents the collapse of a daughter-half during cytokinesis. Assuming a proportional pressure controller, $P_D^{\mathit{div}}=K_d\left(1-V/V_D\right)\text{,}$ the daughter-half bulk modulus $K_d$ determines how strong this mechanical feedback enforces the reference volume $V_D$ of daughter-half D, based on the division plane and mother cell polarization. $P_D^{\mathit{div}}$ is added to the global pressure equation (Eq. S3 in the Supporting material) for each daughter half. Furthermore, the net reaction force due to the pressure difference between “left” (L) and “right” (R) daughter halves, ${\mathit{F}}^r=\underset{S_L}{\int }{P}_L^{\mathit{div}}\mathrm{d}{\mathit{S}}_{\mathit{L}}-\underset{S_R}{\int }{P}_R^{\mathit{div}}\mathrm{d}{\mathit{S}}_{\mathit{R}}\text{,}$ is applied as a reaction force to the furrow ring, ensuring that the global force balance is maintained.

Model parameters

We define the relative mechanical contribution of cortical viscosity compared with mitotic surface tension ${\gamma }_M$ at the division timescale as $\kappa :=\eta R_0/\left(2{\gamma }_M{\tau }_d\right)$. Furthermore, for asymmetric cell divisions, the contribution of $K_d$ in stabilizing cell division is expressed in the relative destabilizing mitotic tension, which we define as the ratio between mitotic tension and the daughter-half bulk modulus: ${\gamma }_n:=2{\gamma }_M/\left(R_0{K}_d\right)$. Finally, the relative contribution of cell-cell and cell-substrate friction is defined as $\lambda :=4\pi \xi R_0^2/\left({\gamma }_M{\tau }_d\right)$. In our model, the viscosity η represents the combined viscosity of the acto-myosin cortex-membrane complex. We assume that η is mostly determined by the long-timescale viscosity of the acto-myosin cortex itself. However, the viscous response is also influenced by internal friction between cortex and cell membrane, ${\xi }_m$, which sets a hydrodynamic length for viscous stress propagation ${\left(\eta t/{\xi }_m\right)}^{1/2}$. In laser ablation experiments in C. elegans, this length scale has been found to be around $14\mu \mathrm{m}$, indicating that cortex viscosity dominates stress propagation at the scale of a cell (50,51). Moreover, an actin-turnover time of ${\tau }_a=24\mathrm{s}$ was observed in C. elegans (51). Since cortical viscosity directly results from turnover of cortical components (50), we obtain for κ an order-of-magnitude estimate as $\kappa \approx R_0{\tau }_a/\left(2t{\tau }_D\right)\approx 2.68$ for P0 in C. elegans, assuming $R_0=17.6\mu \mathrm{m}$ and $t=300\text{nm}$ (see Table 1), which is also consistent with simulations in Menon et al. (52). Furthermore, adhesion is generally low compared with cortical tension during cytokinesis, $\omega \ll {\gamma }_M\text{,}$ and we assume that the stiffness of the contractile ring is dominant compared with the mitotic surface tension, $k\gg 2{\gamma }_M{R}_0$. Values for parameters ${\gamma }_n$ and λ are unknown but are further investigated in this work.

Table 1.

Reference table of division model parameters

Parameter	Symbol	Single Cell Study	P0 C. elegans (ref.)	Units
Cell radius	$R_0$	$10\text{.}$	17.6	$\mu \mathrm{m}$
Effective cortex thickness	t	0.3	0.3	$\mu \mathrm{m}$
Theoretical division duration	${\tau }_d$	300	263	s
Relative mitotic tension	${\gamma }_n=2{\gamma }_M{R}_0^{-1}{K}_d^{-1}$	$0.1-0.8$	$0.1-1.0$$\left(0.125\right)$	$-$
Relative mitotic viscosity	$\kappa =\frac{1}{2}\eta R_0{\gamma }_M^{-1}{\tau }_d^{-1}$	$0.5-40$	$1-10$$\left(1.0\right)$	$-$
Relative contact friction	$\lambda =4\pi \xi R_0^2{\gamma }_M^{-1}{\tau }_d^{-1}$	$0.2-1.7$	$1-100$$\left(1.0\right)$	$-$
Relative furrow stiffness	$k/2{\gamma }_M{R}_0$	$20-160$	$9-90$ (72)	$-$
Transition radius	$R_t$	3.6	3.6	$\mu \mathrm{m}$
Abscission radius	$R_f$	0.8	0.8	$\mu \mathrm{m}$

Model parameters used for the daughter cell shape dynamics and asymmetric division simulations. For c. elegans, the used reference value (ref.) is denoted between brackets. A complete list of used model parameters in the DCM can be found in Table S1 in the Supporting material.

Results

We investigate the cell division model by comparing simulation results to experimental data and theoretical predictions found in literature (6,17,31,36). We first show changes in cell shape dynamics during cytokinesis, after which we investigate furrow ring dynamics, before focusing on spindle positioning and daughter cell volume in asymmetric cell division.

Cell shape dynamics in symmetric cytokinesis

We first investigate cell shape dynamics in symmetric cell division for varying furrow stiffness and mitotic cortex viscosity. We simulated the division of unconfined, spherical cells with different sizes and analyzed the constriction rate and cytokinesis duration. The model reproduces the experimental observations of Carvalho et al. (36) for high relative furrow stiffness $k/2{\gamma }_M{R}_0$, when furrow ring dynamics are effectively prescribed (Fig. S7). As $k/2{\gamma }_M{R}_0$ decreases, tension needs to build up in the ring to achieve similar constriction velocities, resulting in delayed dynamics, and below a critical threshold, failure of furrow ingression; see Figs. S8 and S9 in the Supporting material. Varying cell radius $R_0$, the total duration of cytokinesis ${\tau }_t$ can be estimated from Eq. 1. Setting ${\tau }_d=263\mathrm{s}$ for cytokinesis of C. elegans (36), the experimental fit by Carvalho, ${\tau }_t/{\tau }_d=1+0.386R_t/R_0$, is reproduced for stiff furrows; see Fig. S7. Although the dynamics of a stiff furrow ring are imposed by Eq. 1, the resulting cell shape is nontrivial. Parameter κ, the balance of viscous and active-contractile forces, governs the resulting cell shape. At low κ, daughter cells have a spherical shape, whereas at high κ, cortex viscosity is dominant over surface tension, resulting in oblate spheroidal shapes with the minor axis in the direction of the cell poles (Fig. 2). This is also apparent in the relative pole-to-pole distance $L_p/2R_0$ of the dividing cell. At low κ, $L_p/2R_0$ reaches a steady state at the time of abscission, indicating a spherical shape of daughter cells.

Figure 2.

Cell shape dynamics. (A) Simulation snapshots that show a cross-sectional slice of the cell geometry at different time points during cytokinesis for different values of relative cortex viscosity, κ, and constant ${\gamma }_n=0.48$. The shape of daughter cell halves changes from spherical to nonspherical for increasing κ. (B) Time evolution of contractile ring radius $r_f/R_0$ and pole-to-pole distance $L_p/2R_0$ in function of κ. Pole-to-pole distance increases rapidly at low κ due to the spherical shape of the emerging daughter cells, whereas for high κ, the pole-to-pole distance increase slows down due to the more oblate shapes of the daughter halves. Furrow radius dynamics are unaffected by κ when k was chosen sufficiently high. The vertical dotted line shows the moment of abscission. To see this figure in color, go online.

Volume ratio of daughter halves in asymmetric cytokinesis

Next, we investigate how the volume distribution between daughter halves is affected by mechanical properties and by confinement of the cell. We simulated asymmetric divisions for both confined and unconfined cells and quantified to what extent the simulated daughter cell volume ratio follows the initial volume partition set by the offset of the division plane α. For geometrically confined cells, a distinction was made between parallel ($\parallel$) and perpendicular ($\perp$) confinement, where the orientation angle φ was defined based on the angle between the division plane and confining parallel plates; see Figs. 1 C and 3 A. For all setups, $\beta =2R_0$ represents an unconfined state, whereas for $\beta =R_0\text{,}$ the cells are compressed until the parallel plates are spaced one cell radius apart. The estimated volume ratio for unconfined simulation setups closely follows the initial partition. For decreasing β in the parallel confined configuration, daughter cell asymmetry becomes increasingly sensitive to spindle offset ${\alpha }_n$. The opposite is observed for perpendicularly confined cells, where the volume of daughter cells is less affected by ${\alpha }_n$. This effect can be attributed to the pole-to-pole distance of the cell, which scales as $\sim \beta$ for parallel confined cells and as $\sim 1/\beta$ for perpendicular configurations. We also quantified the effect of mitotic cortical surface tension and viscosity on the volume ratio in asymmetric cell division; see Fig. 3C. In the liquid droplet model of a cell, these properties govern the balloon instability in which asymmetry in daughter cell halves will be further increased, potentially leading to the complete collapse of the smaller daughter half. In our model, this instability is regulated by the relative viscosity of the cortex, κ, as well as by ${\gamma }_n$, the contribution of the mitotic surface tension (driving instability) relative to the daughter half bulk modulus (resisting the instability). As ${\gamma }_n$ increases, mitotic tension becomes dominant over internal rigidity, and cell divisions further increase in asymmetry. For highly asymmetric cell division (large spindle offset ${\alpha }_n$) and large ${\gamma }_n$, this may lead to the complete collapse of a daughter half. The effect of cortical viscosity κ is two-fold. For low ${\gamma }_n$, increased viscosity further increases the asymmetry as it counteracts the mechanical feedback due to internal rigidity. However, at large surface tension and spindle offset, it prevents further a further increase in asymmetry as it slows down the movement of cortical material toward the larger daughter half during the time of cytokinesis. As such, high viscosity rescues cell division and prevents a complete collapse of the daughter half.

Figure 3.

Volume distribution in asymmetric cytokinesis. (A) Simulation snapshots at different time points during cytokinesis for asymetric division. Triangles show the discretization of the cell cortex. Triangles colored in red indicate the furrow ring. In this region, tension is generated, driving furrow ingression. After abscission, the cell is split into two daughter halves, and a new computational mesh is generated. (B) Effect of spindle offset α and orientation of the division plane φ on relative daughter cell volume in a geometrically confined setup where division mechanics dominate over cortex mechanics (${\gamma }_n=0.1$ and $\kappa =0.5$). We compare the volume fraction of the biggest daughter cell to expected volume fraction $V_n=\left(2+3{\alpha }_n-{\alpha }_n^3\right)/4$ given the relative spindle offset ${\alpha }_n=\alpha /R_0$ indicated by the black curve. In this confined setup, two parallel plates with a gap β were added. For confinement parallel to the division plane, $\phi =0$, decreasing gap β increases daughter cell asymmetry. Perpendicular confinement, $\phi =\pi /2$, reverses the observed trend. (C) Effect of mitotic surface tension and relative cortical viscosity on division asymmetry in an unconfined setup. Increasing surface tension destabilizes asymmetric divisions as the pressure difference over the daughter halves increases (bubble-instability), whereas cortex viscosity stabilizes asymmetric divisions at high surface tension and spindle offset ${\alpha }_n$. Cytokinesis fails at high ${\gamma }_n$ and low κ indicated by $V_n=1$ meaning the smallest emerging daughter cell collapses into the largest daughter cell. To see this figure in color, go online.

Effect of spindle positioning on cell volume in early C. elegans development

Finally, we apply our mechanical model for cytokinesis to asymmetric cell divisions that take place during early C. elegans development. We focus on the cleavage phase up to the six-cell stage, as development up to this stage is thought to be predominantly the result of passive cell and cell-cell interactions and less so of active processes, such as cell migration (53) or biochemical queues (54). Consequently, our model assumes cells do not actively migrate or significantly change their mechanical properties—apart from mitotic rounding—and daughter cells inherit their mechanical properties from the mother cell, limiting the scope of our model from the one- to six-cell embryo. We investigated both compressed and uncompressed embryo shapes. Embryo compression occurs naturally in utero, whereas in vitro, it generally occurs as a result of sample preparation for imaging, where the degree of compression depends on the mounting technique used (14). The initial geometry of the compressed embryo was based on segmented embryo shapes obtained from microscopy data (15), whereas the uncompressed shape was estimated by fitting an ellipsoid with the same volume (55). The spindle offset α for each cell division was statistically sampled from experimental data in compressed conditions from Fickentscher and Weiss (6). Furthermore, we assume fixed spindle skew angles φ, with values from experiments of Pimpale et al. (47). In simulations, we assume that cell-cell adhesion is small relative to cortical tension in early C. elegans development (56,57). A full description of the computational model of C. elegans development is provided in the Supporting material. With these parameters, together with well-characterized cell division timings, we simulate the early development of C. elegans up to the six-cell stage. The simulated development reproduces key morphogenic features of early C. elegans development, such as the characteristic diamond configuration in the four-cell stage and dorso-ventral symmetry breaking in the six-cell stage; see Fig. 4 A. The volume of simulated cells was compared with volume estimates based on segmented microscopy images of compressed embryos (15) (Fig. 4 B). We observe that the simulated volume ratio agrees well with experimental observations. Based on this, we conclude that the proposed DCM of cytokinesis captures the dominant mechanical processes that drive asymmetric cell division well. This supports the notion that position α and angle φ of the mitotic spindle are highly robust predictors for volume asymmetry in the C. elegans cleavage phase.

Figure 4.

Simulated volume ratio in C. elegans early development. Volume of daughter cells relative to the volume of the mother cell. Only the volume ratio of the largest daughter cell is reported as the sum of larger and smaller daughter cell is always $\approx$ 1. (A) Simulation snapshots for the two- to six-cell stage as seen in the LR direction (top view) and DV direction (side view). (B) Comparison of estimated cell volume between experiments (15) and simulations for compressed and uncompressed egg geometry. For both compressed and uncompressed simulations, spindle offset α values reported by Weiss et al. in compressed conditions were used. For the P0 division, estimates by Gotta et al. were used (58). Significant differences between simulation and experiment are indicated for a standard two-sided t-test (p-values 0.01 $\ge$^∗$\ge$ 0.001 $\ge$^∗∗$\ge$ 0.0001 $\ge$^∗∗∗). (C) Sensitivity analysis of daughter cell volume, for relative cortex viscosity, surface tension, and contact friction. Black squares mark the reference value of a given parameter. It should be noted that changes in cell-cell connectivity and cell positions were not quantified. To see this figure in color, go online.

We hypothesized that the effect of confinement alone is sufficient to explain the different volume ratio observed in experiments (15). To this end, we adopted spindle positions from experiments in confined conditions but simulated development in an uncompressed egg geometry. However, we observed that simulation results were significantly different from experimental observations; see Fig. 4 B. These results suggest that apart from cell shape changes, egg geometry also influences spindle positioning, as the shape changes alone were found to be insufficient to induce the experimentally observed differences in volume. Since spindle positions seem to be the dominant factor in determining daughter volume ratio, our model suggests that any significant change in the volume of daughter cells should be accompanied by a shift in spindle position. Moreover, taking into account the results presented in the previous section, we expect that the sensitivity of volume ratio to the absolute spindle position depends on the characteristic length scale of the mother cell in the division direction. Hence, this sensitivity increases as development progress and may depend on the specific geometric configuration of a cell as it initiates cytokinesis. In a complementary approach, the model was used to estimate spindle offset α based on experimentally measured cell volume, using a reverse fitting procedure (Supporting material p. 16–17). The results of this procedure are summarized in Table 2. This yields estimates of α for the compressed conditions that agree with experimental measurements (6,58), and it produces a computational prediction for the adjusted spindle offset in uncompressed conditions. In the latter case, the model predicts a mean absolute shift in α of $\pm 300\text{nm}$. The estimated shift is statistically significant (two-sided t-test p $\le$ 0.05) for the AB cell line but not for the P cell line.

Table 2.

Estimated spindle offset α based on linear fitting

Cell	Exp. $\left(\mu \mathrm{m}\right)$	Exp. Source	Compressed Sim. $\left(\mu \mathrm{m}\right)$	Uncompressed Sim. $\left(\mu \mathrm{m}\right)$
P0	$2.630\pm 0.701$	(58)	$2.876\pm 0.104$	$2.549\pm 0.102$
AB	$-.351\pm 0.312$	(6)	$-.394\pm 0.035$	$.088\pm 0.017$
P1	$1.335\pm 0.268$	(6)	$1.393\pm 0.124$	$1.257\pm 0.142$
ABa	$.357\pm 0.278$	(6)	$.287\pm 0.021$	$.527\pm 0.029$
ABp	$-.617\pm 0.255$	(6)	$-.521\pm 0.041$	$-.079\pm 0.032$

Estimates for α used φ data from Pimpale et al. and volume estimates by Thiels et al. as input (15,47). A linear fitting procedure was used where α values were estimated via interpolation (Supporting material p. 16–17).

Finally, a sensitivity analysis was performed with respect to the influence of cortex mechanics on cell division asymmetry in the C. elegans embryo, thereby assessing the mechanical sensitivity of asymmetric cell divisions in confined environments; see Fig. 4 C. To this end, we vary the relative cortical tension ${\gamma }_n$, cortical viscosity κ, and contact friction λ. We observe that most cell divisions are highly robust to changes in cell mechanical properties. However, a high ${\gamma }_n$ was found to influence cell volume, especially for the smaller cells that are characterized by high curvatures, such as ABpl. The effect of relative cortex viscosity κ was found to be negligible, further supporting the notion of increased stability due to geometrical confinement. Similarly, varying the relative contact friction λ produced no significant effects on cell volume ratio. This latter observation is to be expected, as up to the six-cell stage embryo, cell motility is low, and no drastic cell-cell contact area changes occur.

Discussion

In this work, we presented a computational mechanical model of cytokinesis, which explicitly represents the contractile furrow ring that drives ingression of a viscous cortical shell under mitotic tension. We extended the cytokinesis model to account for asymmetric cell division by varying the position and angle of the mitotic spindle complex. In case division mechanics are dominant over cortex mechanics, the model reproduces a simple geometric relationship between spindle offset and daughter volume fraction. Conversely, for large mitotic surface tension, asymmetric cell division may become unstable as the small daughter half collapses in a balloon instability, an effect that in our model is mitigated by cortical viscosity and the internal bulk rigidity. When confined between parallel plates, daughter cell asymmetry increases when the spindle orientation is parallel to the plate and decreases when it is perpendicular to the plate. For uniaxial compression, these differences in sensitivity can be traced back to a change in the characteristic length in the direction perpendicular to the division plane. For nontrivial compressed states, e.g., cells during C. elegans embryogenesis, similar effects occur where the division direction and characteristic length heavily influence the robustness of the system to any perturbation in spindle positioning. For C. elegans specifically, this partially explains the observed changes in volume ratio for ABa and ABp daughter cells.

We demonstrated that the model is able to recapitulate C. elegans embryogenesis up to the six-cell stage, using spindle position and angle as model input. In the case of C. elegans, we conclude that spindle position alone well predicts the observed volume fraction after cell division. This supports the proposition that during cytokinesis, the mechanical properties of the contractile ring are dominant with respect to cortical properties such as mitotic tension and viscosity. In contrast to our initial hypothesis, the model predicts that differences in volume fraction between compressed and uncompressed embryos cannot be attributed solely to the effect of confinement on modeled cytokinesis mechanics alone. Rather, our results suggest that embryo compression also influences the positioning and/or angle of the spindle apparatus, an effect that is not accounted for by our model but may be expected based on cortical flows (10,59). Indeed, in a reverse fitting procedure, we estimated spindle positions based on the experimentally observed volume of cells and predict a difference in spindle position between compressed and uncompressed embryos. The introduction of a nonzero bulk rigidity stabilizes cell divisions in the viscous cell model by preventing bubble-instabilities. Moreover, we predict that additional stability is provided by the eggshell. Indeed, the first division (P0) simulations are highly stable, even at greatly elevated mitotic surface tension. This observation highlights a possible additional role of the eggshell—or similar structures in higher organisms—during the cleavage phase of development. It does not only provide protection against outside chemical and mechanical stress, but it also provides supplementary stability for asymmetric divisions.

The similarity of our model and the visco-active membrane theory of the cortex proposed first by Turlier et al. (22) warrants further discussion. In their model, the formation and constriction dynamics of the furrow ring itself are predicted based on the internal flow and polymerization/depolymerization of cortical material. They proposed that due to accumulation of cortical material near the division plane, viscous stress builds up in the furrow ring to balance line tension, thereby determining the constriction rate of the furrow ring. Furthermore, their model was able to produce the distinct (nonspherical) shape of a dividing sand dollar zygote. In our model, the constriction dynamics are imposed for a sufficiently stiff contractile ring. As such, the emergence or failure of cell division itself is not predicted in our model. Nonetheless, our model does reproduce characteristic nonspherical shapes for high values of $25\le \kappa \le 250$, as were used in the model of Turlier et al. (22). For these high values of κ, our results show that viscosity sufficiently slows down mechanical instabilities in asymmetric cell division to prevent daughter-half collapse. However, for P0 in C. elegans, previous laser ablation experiments suggest that $\kappa \approx 2.68$ (51). For these values of κ, our results suggest that additional stabilization could be furnished by internal rigidity, for example due to the mechanical rigidity of microtubules, but also by external confinement provided by neighboring cells or the eggshell. In this work, we did not investigate the effects of cell-cell adhesion, cell-egg adhesion, and embryo-egg volume ratio, as Giammona et al. have already covered these topics (28). Finally, we also did not investigate the important effect of confinement, cell-cell connectivity, and signaling on spindle offset α and skew angle φ (10,59). Incorporating the underlying mechanisms that mechanically regulate these structures will prove crucial for expanding this model toward fully predictive simulations of development. Nonetheless, this work shows the feasibility for the DCM framework to simulate developing systems such as early C. elegans morphogenesis while allowing for detailed comparison to experimental observations.

Author contributions

M.C. and B.S. conceived the project. M.C., J.V., and B.S. designed and conducted the simulations, M.C. and B.S. performed data analysis. W.T. and R.J. contributed ideas and data for the application to C. elegans embryogenesis. M.C., W.T., R.J., H.R., and B.S. wrote the manuscript.

Editor: Pablo Iglesias.

Supporting citations

References (60,61,62,63,64,65,66,67,68) appear in the Supporting material.

Supporting material

Document S1. Supplementary note

Complete description of the deformable cell model, including validation. Full list of model parameter, sensitivity analysis and reverse fitting procedure for parameter α.