On the embryonic cell division beyond the contractile ring mechanism: experimental and computational investigation of effects of vitelline confinement, temperature and egg size

Image acquisition

Sk embryos were obtained by standard in vitro fertilization protocol (Lowe et al., 2004) using eggs from a single female and sperm form a single male. The temperature was controlled by placing the glass Petri dish on a metal stage with a constant heat exchange by circulating water from a controlled temperature water bath. Over the duration of the experiments, the sample temperature did not deviate from described value by more than 0.2 °C as controlled by a 4,238 Traceable thermocouple thermometer certified to a resolution of 0.1 °C and accuracy of 0.3 °C. The time-lapse imaging at 4 frame/second was done with a regular Nikon camera mounted on a dissection microscope with 10× magnification lens (Fig. 1). The embryos were kept on to ensure normal development. Effects of temperature on cytokinesis are investigated on the basis of time-lapse image series of Sk embryonic cells that exhibit visible polar elongation during the cleavage. To ensure data consistency, only cells that undergo symmetric cleavage in the plane orthogonal to the field of view are preselected for subsequent image analysis. Out-of-plane and non-symmetrically dividing cells are excluded from analysis.

Image processing

3D (2D + time) stacks of each experimental time series of images are denoised and semi-automatically segmented with the help of Amira v4.1 (Mercury Computer Systems, Arlington, VA, USA), see Fig. 2A. Subsequently, spatial–temporal isosurfaces (Fig. 2B) and contours (Fig. 2C) of dividing cells are generated for all previously segmented cells using Amira’s surface and contour generating routines. For all cells and all time steps s = 1..N, coordinates of mass center points and the lengths of the shortest (equatorial furrow) F(s) and longest (polar) L(s) embryonic axes are calculated using a C computer program developed in-house (Fig. 2D). Time-series of F(s) and L(s) are smoothed using the 5-point masked median filter for subsequent computation of time-derivatives (F′(s), L′(s)) and detection of time steps of the image sequence s ∈ [s_s, s_e] corresponding to the first embryonic cell division. For this purpose, the sum of absolute derivatives (SAD = |F′(s)| + |L′(s)|) is calculated which serves as an indicator of local curve steepness (Fig. 3). Start (s_s) and end (s_e) time points of the first embryonic cleavage are determined automatically as arguments of local minima left and right from the absolute maximum of SAD and subsequently validated by visual inspection. For an invariant description of cytokinesis, a dimensionless time t = (s − s_s)/(s_e − s_s) ∈ [0, 1] is introduced.

Geometrical modeling

For simulating cellular deformations during the cleavage, a spherically-shaped 3D triangulated surface model is generated. The closed surface is filled up with an unstructured tetrahedral grid using Amira’s TetraGen tool. For every new step of the multi-step simulation procedure, the surface generation and tetrahedral grid generation is repeated anew.

Dimensions of all essential geometrical parameters required for simulation of cell division (such as cell cross-section, furrow width, spindle length, etc.) are converted from the physical scale (i.e., in μm) into the dimensionless scale of the virtual cell model according to the relative proportions, see example in Table 1.

Physical modeling

Following the assumption of the contractile ring theory, we initially model the first embryonic cell division as a deformation of a three-dimensional elastic ball successively constricted in its equatorial plane by the contracting cleavage furrow. Starting from this one-material, one-mechanism model, we iteratively refine and extend it by minimizing the deviation between computationally predicted and experimentally observed cell shape changes.

Cellular matter is approximated as an elastic (Hookean) material described by the piecewise linear stress–strain relationship (St. Venant-Kirchhoff material law) (Ciarlet, 1988): (1)${\displaystyle \mathbf{\sigma }\left(\mathbf{\epsilon }\right)=\frac{E}{1+\nu }\left(\mathbf{\epsilon }+\frac{\nu }{1-2\nu }\text{tr}\left(\mathbf{\epsilon }\right)\mathbf{I}\right),}$ where σ denotes the Cauchy stress tensor, ε is the Green–Lagrange strain tensor and (E, ν) are the Young’s modulus and the Poisson’s ratio, respectively. The linear stress–strain is a basic property of mechanical continuum in the range of small relative deformations, i.e., ε ≤ 0.05. However, due to the fact that the strain tensor is a nonlinear function of displacement ε(u): (2)${\displaystyle \mathbf{\epsilon }\left(\mathbf{u}\right)=\frac{1}{2}\hspace{0.17em}\left(\mathbf{\nabla }{\mathbf{u}}^{\mathrm{T}}+\mathbf{\nabla }\mathbf{u}+\mathbf{\nabla }{\mathbf{u}}^{\mathrm{T}}\mathbf{\nabla }\mathbf{u}\right)}$ the equations of elasticity theory are, in general, nonlinear. In the range of small deformations, the higher order quadratic terms in (2) are usually omitted resulting a fully linear formulation of continuum mechanics with respect to the displacement.

The deformation of the entire cell is calculated numerically by integrating the partial differential equations of elastostatic equilibrium (3)${\displaystyle \text{div}\hspace{0.17em}\mathbf{\sigma }\left(\mathbf{\epsilon }\right)+\mathbf{f}=0}$ under consideration of the boundary conditions and forces f that are given implicitly in the form of predefined boundary displacements, i.e., radial contraction of the equatorial cleavage furrow region. To solve the boundary value problem given by (1)–(3) for a 3D spatial domain, the Finite Element Method is applied in a way as previously described (Gladilin et al., 2007).

The above model of cell mechanics does not consider temperature as an explicit material parameter. We do not strive for formulation of a constitutive law with an explicit temperature dependency, because differently from classical engineering materials, temperature affects not only passive material properties of living cells, but also rates of all chemical reactions, which, in turn, may have an indirect impact on material cell behavior. Instead, we consider temperature as an implicit parameter which according to the literature (Chan et al., 2014) can be expected to influence canonic material parameters (i.e., cell stiffness).

Effects of vitelline confinement on cytokinesis: plausibility simulation

Evidently, changes of the cellular shape during the embryonic cleavage are mechanically restricted by the vitelline membrane, see Fig. 4. Relatively loose vitelline confinement let dividing Sk cells freely expand along their polar axis (Fig. 4A), while Xl cells are bound to deform within a tight vitelline confinement during the entire cleavage (Fig. 4C). We wanted to understand how differences in boundary constraints affect mechanics of cell division. To this end, the boundary conditions corresponding to Sk and Xl type of vitelline confinement need to be appropriately incorporated into mechanical cell model. From the viewpoint of structural mechanics, boundary conditions on the outer surface of dividing Sk cells represent an example of the Neumann type of (free) boundary, while vitelline confinement of Xl cells can be described by the slippery (sliding) boundary condition which allows only tangential displacement along the boundary surface and forces its normal component to vanish, i.e., u n = 0. Figure 4B shows three characteristic exemplary steps in the Finite Element simulation of the cell cleavage for these two, considerably different types of boundary conditions, see the corresponding video sequences (Videos S1 and S2) In both cases, an incompressible one-material linear elastic model of cellular matter is assumed and the boundary loads are given implicitly by successive contraction of the equatorial furrow region with the width indicated in Table 1. To minimize the error due to linear elastic approximation, the FE simulation of cell cleavage is performed in 10 successive steps, by each of which a relatively small rate of 5% furrow contraction is applied. For each next step of the FE simulation, the deformed cellular surface from the previous iteration is used to regenerate a consistent, high-quality tetrahedral mesh and to avoid undesirable artifacts due to largely deformed and deteriorated tetrahedrons.

In addition to striking differences between the geometry of vitelline-confined and vitelline-unconfined cell deformations, significant differences in the amount of mechanical energy required for these two types of cellular division can be expected. For this purpose, the volume integral (i.e., sum over tetrahedral elements) of the strain energy density is calculated (Ogden, 1984) (4)${\displaystyle W=\int \left[\mathbf{\sigma }\hspace{0.17em}\mathbf{\epsilon }\right]dV=\int \left[\frac{\lambda }{2}\mathbf{(}\text{tr}\mathbf{\epsilon }{\mathbf{)}}^2+\mu \text{tr}\mathbf{(}{\mathbf{\epsilon }}^2\mathbf{)}\right]dV}$ where $\lambda =\frac{E\nu }{\left(1+\nu \right)\left(1-2\nu \right)}$ and $\mu =\frac{E}{2\left(1+\nu \right)}$ are the Lame constants. For incompressible material (i.e., ν = 0.5), the first term in (4) is neglected. Our numerical simulations show that cleavage of a vitelline-confined cell requires 4.8 times higher mechanical energy in comparison to the vitelline-unconfined case provided all other cell parameters (including cell size/volume) in both cases are equal.

As a result of tight vitelline confinement, the deformation of Xl cells is directed inwards and practically not visible in the outer cell contours. Therefore, all further experiments and model validations are performed using image data of Sk cells that are not constrained by vitelline envelope most time of the cleavage. By the end of cleavage, some elongated Sk cells can, however, come in contact with vitelline membrane, which restricts their further polar expansion. Due to correlation between vitelline cross-section and temperature, which will be discussed in more details in subsequent sections, the effects of vitelline as a mechanical obstacle are more frequent by low temperature conditions.

Effects of furrow width and stiffness inhomogeneity: model sensitivity analysis

Geometrical and physical parameters of embryonic egg cells can considerably vary from species to species as well as during the cleavage time for the same egg cell. We want to estimate sensitivity of our computational predictions with respect to polar cell elongation in dependency on variable furrow width and cell stiffness that represent two key parameters of our contractile ring model of the cell cleavage. Previous works (Schroeder, 1972; Zang et al., 1997; Foe & Von Dassow, 2008) describe large changes of the furrow width (i.e., 3–17 µm) during the first embryonic cell division of different species. Remarkably, all authors make a similar observation that the furrow width reduces from the maximum value in the first half of the cleavage to the minimum value which is reached by the end of cytokinesis. Inhomogeneity of stiffness between equatorial and polar regions has been contradictory discussed in the past. While Matzke, Jacobson & Radmacher (2001) reports increase of stiffness in equatorial furrow region, Koyama et al. (2012) makes completely opposite statements based on their image analysis and knock-down experiments. To analyze the effects of furrow width and stiffness inhomogeneity, we numerically simulate the cellular elongation resulting from the initial 10% contraction of the cleavage furrow. The computational simulation is performed for a range of reasonable values of the furrow width (4–16 μm) and stiffness gradient between equatorial (E_e) and polar (E_p) cell regions (log(E_e/E_p) ∈ [ − 2, 2]). The result of these simulations are shown in Fig. 5 allow the following conclusions:

• The relative cell elongation triggered by the initial 10% contraction of the cleavage furrow varies between 2.5% and 6.2% within the entire range of reasonable values of furrow width and equatorial-to-polar stiffness gradient.

• Increase in furrow width results in increased polar cell elongation under otherwise same conditions.

• Increased stiffness of equatorial region (i.e., E_e/E_p > 1) leads to slightly larger polar elongation in comparison to the homogeneous case E_e/E_p = 1; softening of equatorial furrow region (i.e., E_e/E_p < 1) has the opposite effect.

• Twofold increase of furrow width has 10 times higher effect on polar cell elongation as twofold difference between equatorial and polar cell regions.

• Predictions of cellular elongation are, in general, not unique with respect to combination of model parameters.

Effects of temperature on cytokinesis: experimental data vs. simulation

Altogether, from two Sk populations exposed to two different temperatures (18 °C and 26 °C), 10 and 16 cells are respectively selected for image analysis which is performed as described above, including segmentation of ROI, extraction of cellular contours and determination of the equatorial furrow F(t) and embryonic polar L(t) lengths for all time steps (t ∈ [0, 1]) corresponding to the first cleavage. For invariant representation of data, absolute lengths are substituted by dimensionless normalized values (F_n(t) = F(t)/F(0) − 1 and L_n(t) = L(t)/L(0) − 1) that have been used for quantitative description of cytokinesis in previous studies (Hiramoto, 1958; Pujara & Lardner, 1979; Akkas, 1981). Since dividing cells do not separate during the cleavage sufficiently enough to accurately detect their boundaries in the furrow region, the cross-section of equatorial contractile ring F_n(t) is not precisely quantifiable for the entire duration of the cleavage. Consequently, the dimensionless cleavage time (t ∈ [0, 1]) is calculated from the time course of image frames as described in ‘Image processing’ and the polar embryonic length L_n(t) is used for quantitative description of cell elongation. Figure 6 shows comparison of our measurements of polar elongation of dividing Sk eggs under two different temperature conditions (18 °C, 26 °C) vs. FE simulation of vitelline- and spindle-free contractile ring model of egg cleavage under assumption of the constant furrow width (FW = 8 µm or 2% of the initial cell cross-section). Comparison of experimental data and computational simulation of the Sk egg cleavage shows that:

• Cell elongations L_n(t) for 18 °C and 26 °C do not exhibit statistically significant difference when compared as a whole (p-value: 0.88),

• At the end of the cleavage (t ≥ 0.7), elongation of 18 °C Sk cells is significantly slower in comparison to 26 °C probe (p-value: 0.028),

• The result of our FE simulation matches well with experimental observations for both temperature conditions (p-values: 0.790 (18 °C) and 0.685 (26 °C), respectively),

• The linear pattern of simulated cell elongation differs from experimentally observed L_n(t) curves that exhibit slightly nonlinear behavior.

Effects of spindle elongation in small cells: extension of contractile ring model

The maximum elongation of Sk egg cells by the end of the cleavage amounts in average to 30.9%. Our computational simulation based on the contractile ring model with the unconstrained cell boundaries and constant furrow width predicts a similar elongation magnitude of 30%. However, significantly larger elongation rates of embryonic eggs have been reported for other species, i.e., 43.8% for Cj (Hiramoto, 1958), and previous theoretical models fail to explain such a large elongation of dividing cells on the basis of differences in passive material properties (Pujara & Lardner, 1979; Akkas, 1981; He & Dembo, 1997). As we have seen above, larger polar elongation can be theoretically attributed to larger width of the cleavage furrow. However, a strongly nonlinear pattern of Cj elongation reported in Hiramoto (1958) cannot be explained by larger constant or variable furrow width alone. Other sources of mechanical forces that are capable to drive such a large cell polarization has to be considered in addition to the contractile ring mechanism. Mitotic spindle elongation represents such an additional mechanism that is known to contribute to mechanics and shape changes of dividing embryonic cells (Hiramoto, 1956; Schroeder, 1972; Wühr et al., 2009). The reason why mitotic spindle can be expected to contribute to elongation of Cj to a significantly larger extent than in Sk cells, we see in different relative proportions between the cell size and the maximum spindle length which has been shown to have a principle upper length limit of 60 µm (Wühr et al., 2008; Dumont & Mitchison, 2009). Figure 7 demonstrates different relative proportions of the maximum spindle length to typical cross-sections of Sk (400 µm) and Cj (100 µm). Since small variations in spindle length and furrow width may have a large impact of elongation of small Cj eggs, they are both considered to be a variable parameters of our computational model that approximate ranges are estimated on the basis of available literature values (Schroeder, 1972; Zang et al., 1997; Foe & Von Dassow, 2008). Figure 8 shows the result of a stepwise multiparameteric fit of our FE model to the Cj elongation data (Hiramoto, 1958) in comparison to cumulative measurements and simulation of Sk embryonic cells. From the best fit between computational modeling and experimental Cj data, our simulation predicts successive decrease of the furrow width from 13 µm down to 3.6 µm accomplished by slightly nonlinear elongation of mitotic spindle from 30 µm to 53.6 µm, see Fig. 9. Tabulated values of the relative egg elongation of all experimental and simulated data as well as fitting parameters as a function of dimensionless cleavage time are summarized in Tables 2 and 3, respectively.

Effects of temperature on vitelline membrane

As we have seen above (Fig. 6), Sk cells exposed to lower temperature 18 °C experience an unexpected slow-down of the elongation rate at the end of the cleavage (t = 0.8, 0.9, 1.0) which is present neither in the experimental 26 °C probe nor in computational simulation. From the viewpoint of continuum mechanics, such an abrupt stop of polar cell expansion can be principally attributed to (i) a sudden decrease of the effective stretching force and/or (ii) mechanical obstacle which physically restricts free deformation of cellular matter. Both mechanisms can not be generally excluded. Contractile activity of actomyosin complex has been observed in the second phase of cleavage not only in the equatorial furrow but also in polar regions (Foe & Von Dassow, 2008) which can effectively decelarate cell polarization. However, the fact that our Sk cells are surrounded by the vitelline envelope let us firstly suspect the cause of abruptly reduced cell elongation in mechanical restriction of cell deformation by vitelline. Careful analysis of dynamic changes in vitelline membrane cross-section during the cleavage provides evidence for this assumption. Figure 10 shows time cources of the vitelline membrane cross-section to polar cell length ratio (V/C) of 18 °C and 26 °C Sk cells. Surprisingly, the relative cross-section and dynamics of expansion of vitelline envelope during the cleavage strongly depends on temperature. As one may see, Sk cells (n = 23) exposed to lower temperature (18 °C) exhibit significantly more narrow vitelline envelope (median(V/C) = 1.32) already at the begin of the cleavage (t = 0) compared to population of cells (n = 32) observed by 26 °C (median(V/C) = 1.48). In the late phase of cleavage (t ≥ 0.7), the vitelline envelope of 18 °C incubated Sk cells becomes a mechanical obstacle for polar egg elongation (median(V/C) = 1.0). In contrast, 26 °C Sk embryos do not experience such a boundary constraint (median(V/C) > 1) and freely elongate up to the cleavage end. Comparison of the entire time course of V/C ratios shows a statistically significant difference between measured populations of 18 °C and 26 °C incubated Sk cells (p-value: 0.009).