Cells undergo traction-mediated cytofission

We next sought to determine whether our model cells could undergo traction-mediated cytofission, a process whereby multinucleated cells can divide during interphase [15]. We incorporated adhesion into the model taking advantage of recent measurements of the traction experienced by motile Dictyostelium cells (Fig. 1C) [16]. Starting from a spherical cell, we applied protrusive forces in directions 180° apart (Fig. 1D). Though this assumption represents a geometrical simplification that allows us to take advantage of cylindrical symmetry, the amount of force is proportional to the cross-sectional area of the cell (initially a circle) and is representative of the protrusive force experienced by a cell that makes a hemispherical contact with the substrate. This force led to relatively slow cell elongation and initially, concomitant slow furrow ingression (Fig. 2B; Video S1). However, as the furrow narrowed, the cortical tension combined with an increase in local curvature to amplify the local stress. This, in turn, accelerated the rate of furrow ingression, increasing the local curvature further. This positive feedback loop caused a drastic pinching of the furrow, leading to daughter cell separation (Fig. 2B,C). It must be noted that the mean curvature depends on the 3-D nature of the local geometry which involves both axial and radial components. The former is decreasing as the furrow ingresses, but the latter increases greatly during constriction.

Experimentally, it is documented that separate molecular mechanisms are needed to promote the scission of the bridge joining the two daughter cells [17], [18]. Furthermore, measurements of the furrow ingression dynamics show the existence of a bridge-dwelling step that is quantitatively separable from the mechanical stresses that drive furrow ingression [10]. For these reasons, we did not attempt to simulate the final bridge severing and stopped the simulations at this point.

Spatial heterogeneities in cortical tension can initiate cell division, but only in adherent cells

The rapid rate at which curvature-induced differences in cortical tension enabled furrow ingression in the previous simulation led us to posit whether spatial differences in the material properties of the cell could initiate ingression and eventually give rise to sufficient forces leading to cell division. Using micropipette aspiration, we previously measured the effective cortical tension under several contrasting conditions, including interphase vs. mitotic, WT vs. myoII null, and furrow vs. polar regions and demonstrated that the furrow exhibits a 20–30% higher effective cortical tension relative to the poles [8], [12]. We incorporated this heterogeneity into the model and simulated cytokinesis in non-adherent (Fig. 3A) and adherent conditions (Fig. 3B; Fig. S5; Video S2). In both cases, heterogeneity in effective cortical tension and the resultant difference in Laplace-like pressures cause furrow ingression. In non-adherent cells, however, furrow ingression stops shortly after commencing and is not sufficient to cause further ingression or cell division. By increasing the difference in effective cortical tension, we were able to achieve cell division, but this required non-physiological differences (3–10 fold) in effective cortical tension between pole and equator (not shown). On the other hand, the addition of transient adhesive and protrusive forces led to successful cell division (Fig. 3B). These forces appear to be required to induce a sufficient change in morphology (specifically, curvature) from which cortical tension can complete furrow ingression.

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Figure 3. Simulation of myoII null cells.

Morphological changes in a model where there is a spatial difference in cortical tension for both non-adherent (A) and adherent (B) cells (Video S2). Simulation times are from the initial spherical shape. The distribution of the stresses in the adherent case is shown in Fig. S5. C. Experimental data are taken from myoII null cells dividing on a surface. Experimental times are from Video S3. Scale bar denotes 10 µm. D. Comparison of the furrow thinning trajectory. The experimental data represents mean ± SEM and are taken from reference [12]. To compare the shapes at comparable times, time is rescaled so that the cross-over points coincided (Methods). E. Curvature in a simulation of adherent cells. The curvature at the furrow initially decreases slowly but reaches a minimum before increasing. This causes the stress to increase further increasing curvature and thereby closing a positive feedback loop which leads to rapid cell ingression. The curvature of the daughter cell changes relatively little.

https://doi.org/10.1371/journal.pcbi.1002467.g003

It is well documented that Dictyostelium cells lacking functional myosin II cannot divide in suspension, but successfully divide when placed on an adhesive surface [19]; similar observations have been made of mammalian cell culture cells [20] (Fig. 3C). Though this division is similar to those observed in WT cells, there are some significant differences. The furrow ingression dynamics (quantified as the time-dependent change in the relative furrow diameter) display biphasic behavior, in which a slow phase of ingression is followed by a rapid one [10]. We found strong agreement between the furrow-thinning dynamics predicted by our simulation and those measured experimentally in myoII null cells (Fig. 3D; Video S3). Plotting the curvature at furrow and poles during division, it is clear that the second rapid phase of furrow ingression can be attributed to the large increase in force that comes from an increase in mean curvature at the furrow (Fig. 3E) as the radial component of curvature begins to dominate. There are some noticeable differences in the shapes of the simulated cells when compared to the myoII null cells (Fig. 3C,D). In real cells, protrusions are more “stochastic” causing ruffling at the poles. In our model, protrusive stresses are applied uniformly across the boundary and lead to a rounded shape. The treatment of adhesions is also likely to cause some of these differences. In our model, adhesion is modeled as a homogeneous friction, whereas in cells it is more likely to be localized, and this will affect the shape [21]. Furthermore, in myoII null cells, cortexillin I is not as focused in the cleavage furrow as in wild-type cells [22], [23], which could broaden the zone of increased elasticity

Contractile force from myosin II can also drive furrow ingression

Having established that material heterogeneities cannot initiate division but can provide the required force to finish it, we next considered the effect of a myosin II contractile force in our simulations. To this end, we determined the location of myosin II motors from fluorescent images of GFP-myosin II (Fig. S1) and distributed a contractile force temporally and spatially based on the measured distribution of myosin II motors in the cortex (Methods). Incorporating this contractile force in simulations of non-adherent cells led to successful division (Fig. 4A; Video S4). This demonstrates that a cell in suspension can initiate division by substituting the initial ingression provided by adhesion and protrusion on surfaces by myosin II constriction at the furrow. We also observed division in simulations of adherent cells (Fig. 4B; Video S5). Interestingly, cells that are adherent but do not apply protrusive forces did not divide successfully in simulation (Fig. S2). This suggests that the primary advantage of the adherent surface is that it enables cells to apply protrusive forces. Without these, adhesion acts to resist the myosin II forces and prevent sufficient cellular deformation that would otherwise enable cell division to proceed successfully. Defective cytokinesis on adherent surfaces has been documented in several Dictyostelium strains that have aberrant actin polymerization. In cells lacking coronin, an actin binding protein, attachment to the surface does not facilitate cell division [24]. Similarly, cells lacking AbiA, a component of the SCAR complex, exhibit deficient cytokinesis in adherent conditions [25].

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Figure 4. Cell division in the presence of a contractile force.

Simulation of dividing cells in both non-adherent (A; Video S4) and adherent conditions (B; Video S5). In the latter we also considered the effect of strain-stiffening as defined by Equation 11 (C; Video S6). Simulation times are from the initial spherical shape. D. Experimental comparison is with WT cells. Experimental times are from Video S7. Scale bar denotes 10 µm. E. Comparison of furrow thinning trajectory. Experimental data represent the mean ± SEM and are taken from reference [12]. We rescaled the time axis to compare the shapes at comparable times, by shifting the time so that the cross-over times are denoted as 0 s (Methods). The elapsed time between the start of the simulation and the cross-over time for each simulation is given in the legend. F. Pole-to-pole distance as a function of time.

https://doi.org/10.1371/journal.pcbi.1002467.g004

Beyond the cell's ability to divide in non-adherent conditions, these simulations show some further differences from those of myoII null cells. The initial rate of furrow ingression in these simulations is faster than observed in the simulations devoid of myosin II contractile force. This is expected as the initial deformation now includes the cooperative interaction of two force generating subsystems. Differences are also seen in the shape of the daughter cells, as these simulations give rise to rounder cells than cells from simulations that lack myosin II contractile forces. These observations are in agreement with experimentally measured differences between WT and myoII null cells (Fig. 4B vs. 3B) [10].

Strain-stiffening slows down division

Comparing the simulated furrow-thinning trajectory to that measured experimentally in WT cells did reveal some important differences (Figs. 3, 4). The furrows in our simulations exhibit the same sharp drop in radius that is seen in our models of myoII null cells, which can be attributed to the large rise in pressure due to the increase in curvature. This sharp drop-off, which is not seen experimentally, leads to faster division than in real cells. To account for this difference we considered the possible role that strain-stiffening may have on furrow ingression. Strain-stiffening is a non-linear effect whereby materials harden when deformed sufficiently; this has been observed in several biopolymers [26]. Hallmarks of strain-stiffening can be seen in other aspects of Dictyostelium cellular and cytokinesis mechanics in a myosin II-dependent manner. For example, in response to pressure jumps from micropipette aspiration, cells missing myosin II show non-linear effects that are absent in WT cells, suggesting that myosin II pre-stresses the network, leading to strain-stiffening [12]. We incorporated a phenomenological description of strain-stiffening into our model (Methods) and simulated the system. As expected, the initial rate of furrow ingression was unaffected. However, as the furrow diameter became small enough to cause strain-stiffening, the furrow ingressed more slowly, matching the rates observed experimentally (Fig. 4C–E; Videos S6 and S7). While strain stiffening slows down the cytokinetic progression of WT strains, we have not observed this slowdown in experiments of myoII null cells. This suggests that myosin II is a fundamental component that provides this stiffening effect, an observation that is consistent with our measured material properties of myoII null cells [8], [12].

Using this full model we considered the effect that changing the material properties of the cell have on the furrow ingression dynamics. For example, we varied the parameter controlling elasticity (K in Fig. 1B, according to Equation 11) and simulated furrow ingression (Fig. S3). Increasing the elasticity constant by 40% led to a slower, more linear initial ingression (cross-over time increased from 370 to 420 s), as well as slower division overall (415 to 495 s). In contrast, decreasing the elastic constant 30% shortened the cross-over time (370 to 350 s) as well as the total trajectory (415 to 380 s). The simulated trajectories of the model with reduced elasticity are reminiscent of experiments of cells lacking globally-distributed proteins, such as RacE and dynacortin, that have a strong effect on the viscoelastic moduli and act to slow furrow ingression [10].

Finally, the model allows us to sort out an additional point about cytokinesis furrow ingression dynamics. In particular, it is often thought that myoII null cells divide by simply crawling apart. However, our simulations indicate key differences in mitotic cell division for both WT (Fig. 4B,C) and myoII null cells (Fig. 3B) and interphase traction-mediated cytofission (Fig. 2B). By plotting the pole-to-pole distance as a function of time (Fig. 4F), it can be seen that interphase cells drive fission solely by crawling apart. This leads to significant pole separation as well as long and thin morphologies (Fig. 2B). In contrast, mitotic cells that have spatial heterogeneity in their mechanical properties initiate division through protrusion, but divide quite differently, with pole-to-pole distances that are similar to WT cells.

Level set method

The level set method takes an Eulerian approach, tracking a moving boundary (denoted Γ(t)) on a static Cartesian grid deformed by a continuum stress field across the simulation domain [13]. In our simulations, we take a two-dimensional domain and assume cylindrical symmetry about the division axis (Fig. 1A). The level set formalism defines a potential function φ(x,t) for which the boundary is the zero-level set: Γ(t) = {x∈R² | φ(x,t) = 0}. In our simulations, we initialize the potential function with the signed distance function, whose magnitude equals the shortest distance from a point x∈R² to the curve Γ(t) and whose sign is positive if the point is outside the cell and negative otherwise. In practice, as the potential function evolves over time, it can become quite steep or flat, leading to numerical errors. These can be minimized by re-initializing the potential function periodically using the equation(1)where S(φ(x,0)) is taken as +1 inside the cell, −1 outside the cell and zero on the cell membrane.

The potential function evolves according to the Hamilton-Jacobi equation(2)The vector v(x,t) is the velocity of the level set moving in the outward normal direction which, in our simulations, describes the cell's membrane protrusion and retraction velocities. These are driven by a combination of active and passive stresses acting on a mechanical model of the cell, to be described next.

Mechanical model

Previously we developed a mechanical description of a cell in the level set framework and fitted a viscoelastic model topology with parameters obtained from measurements of cells deformed using micropipette aspiration [14]. The model assumes that the cell deformation obeys , where v is the velocity defined above, and x_m is the displacement of the membrane (Fig. 1A, B). The total membrane displacement is the sum of the displacements of the cortex (x_cor) and cytoplasm (x_cyt). To describe how stresses affect these, we use a Voigt model, which consists of the parallel connection of elastic (K) and viscous (D) elements, to represent the cortex connecting the cell membrane and the cytoplasm (Fig. 1B). The viscous component describes the association and dissociation dynamics of actin cross-linkers. The cytoplasm is modeled by a purely viscous element (B) placed in series with the Voigt element. In our simulations, we use stress rather than force to drive the cellular deformations thus accounting for the extra µm² found in the parameters in our model. The model assumes that these displacements occur normal to the cell surface. This neglects bending effects, which are relevant at much smaller length-scales than those we consider in modeling cytokinesis [27], [31].

In the simulations, the total stress (σ_tot) is applied at the cell boundary, according to: , where x_cor and x_cyt represent the positions of the cortex and cytoplasm, respectively. Using the membrane displacement, x_m = x_cor+x_cyt, we can rewrite the system of equations as(3)We thus obtain the membrane velocity solving first for x_cor and then for . This value is entered into the Hamilton-Jacobi Equation (Eqn. 2).

Stresses acting on the cell

The total net stress (σ_tot) is computed for the simulation domain as the vector sum of all stresses acting on the cell. This includes stress contributions from active components, adhesion (σ_adh), protrusion (σ_pro), and myosin-based contraction (σ_myo), as well as passive components due to surface tension (σ_ten) and volume regulation (σ_vol). Thus(4)These individual components are now described in detail.

Adhesion model

Our model of adhesion uses a continuum stress field to counteract cellular deformations [32] and is based on defining an adhesion map, as previously described [33]. Though our simulations assume that the cell has cylindrical symmetry, for the purposes of computing adhesion and protrusion, we instead consider the cross-sectional area in the (z,r) plane, which more closely corresponds to the contact area between cell and substrate. We compute area densities in both r and z directions, normalized to the total cell cross-sectional area:(5)Here 1(z,r) is the indicator function that equals one when the point (z,r) is inside the cell and zero otherwise, and the summations are done over all simulation points in either the z- or r-direction (Fig. 1C). These densities describe the fraction of the cell-substrate contact area that lie in the respective strips either in the z- or r-directions. We multiply these two densities and scale by the maximum adhesion stress (σ_adh-max) to generate a spatial adhesion map:(6)This adhesion is applied spatially as a resistive stress element that counteracts the net effect of the other stresses. To evaluate this new model we simulated the cellular response to a series of pulses (Fig. S4) and compared this response to that of the nominal model. Simulations that incorporate adhesion show a delayed initial response and these cells also take longer to reach steady state.

Protrusion model

We incorporate protrusive forces based on several assumptions (Fig. 1D). First, protrusion acts at both ends of the cell to drive the cell apart. Thus, the protrusive stress acts away from the z = 0 line. Second, the local protrusive forces depend on the contact area (as calculated by D_r(z) above) and increase as you move away from the cleavage furrow (scaled by a linear function l(z) with values of zero at the center of the division axis (z = 0) and one at the poles). Finally, the protrusive force decreases over time as the cell is dividing. We incorporate this by including an exponential function indexed by the furrow diameter (w_f(t), defined as the diameter of the cell at the midpoint along the z-axis). Together, these assumptions lead to a protrusion stress whose magnitude is given by(7)where σ_pro-max is maximum stress applied (Table 2). Though the stress is assumed to act along the z-axis (Fig. 1D), only the component normal to the surface is used in the simulations. The model used here is phenomenological, but captures the net movement of the membrane away from the division plane. Other approaches, which look at finer scale effects for modeling protrusion, have been considered in the literature [34], [35].

Myosin II contractile force

An active contractile force from the work of myosin II against the cytoskeleton is present in wild type cells. This force acts tangentially to the cortex, thereby constricting the cell and, because we assume cylindrical symmetry, this reduces the circumference (Fig. 1E). This has the net effect of reducing the furrow diameter (with a stress reduced by a factor of 2π to account for conversion from circumference to radius). Thus, to incorporate this into our model, we assume that the contractile stress acts radially inward. The magnitude of the local force depends on two things, the maximum stress generated by myosin II and the local distribution of myosin II.

To compute the maximum stress we note that if we assume 3.4 µM total cellular concentration of myosin II monomers (each monomer is composed of two heavy chains, two essential light chains and two regulatory light chains) [11], [36], then a mitotic cell with a radius of 5 µm contains 1×10⁶ myosin monomers (2×10⁶ heads). Given the Dictyostelium myosin II unloaded duty ratio (0.6%) and the force generated by the power stroke of the myosin (3 pN), the maximum total force that can be generated from myosin II is ∼40 nN, assuming no load-dependent shifts in the duty ratio. Because only ∼20% of the myosin II is found in the assembled bipolar thick filament state, most of which resides in the cortex [11], [37], the resulting maximal force is 10 nN. This number is used to compute the total maximum stress by dividing by the cellular area (4πR²).

To apportion this stress spatially, we imaged myoII::GFP-myoII cells (mhcA (HS1):: pBIG:GFP-myosin II; pDRH:RFP-tubulin) undergoing cytokinesis as previously described [38]. From this movie, the GFP-myosin II fluorescent intensities were extracted to quantify myosin density. Cell images were aligned by their centroids and along the division axis. For each image, edge detection was performed to identify cell periphery. Using this edge, the GFP-myosin II intensity was computed for 5 pixels (1 µm) inwardly normal from the boundary, a region likely to contain cortical myosin. An average of these intensities was assigned as the local myosin density at that boundary point. The cell shape was averaged across both its axes of symmetry along with the GFP-myosin II distributions to construct a symmetric myosin profile along the division axis. This profile was smoothed using a cubic smoothing spline. For each image in the time series, a one-dimensional profile was constructed, indexed to the position along the division axis and the measured furrow diameter (Fig. S1). The resultant map (myo(r,z)) describes the distribution of myosin as a function of radius and is used to generate a stress:(8)where n is the outward normal unit vector.

Surface tension

Local differences in mean curvature and surface tension give rise to spatially heterogeneous stresses on the cell. The stress differential across the boundary, described by the Young-Laplace relationship, is given by σ_ten = γ(z)κ_mean(z)n, where γ(z) describes the local cortical tension, κ_mean is the mean curvature and n is a normal unit vector.

The mean curvature, κ_mean, is the arithmetic mean of two principal curvatures (κ_mean = ½(κ_2D+κ_P)) [39]. The first is computed using a Lagrangian formulation based on the cellular boundary: κ_2D(x, y) = (x′y″−y′ x″)/(x′²+y′²)^3/2 where the point (x,y)∈Γ. The primes denote spatial derivatives along the boundary and are approximated by the center weighted difference between two points [13]. The computation of the second principal curvature takes advantage of the cell's cylindrical symmetry: κ_P = N_r(r)/r, where N_r(r) is the normal in the radial direction at a given point, and r is the radius of the cell at that location [39].

For interphase cells, we assume that cortical tension is homogeneous around the cell with a nominal value of 1 nN/µm [10]. For mitotic myoII null cells, we assume a spatially heterogeneous γ with values of 0.5 and 1.0 nN/µm at the pole and furrow, respectively [8], [12]. We interpolate these values using a Gaussian profile:(9)where R₀ is the initial radius of the cell and z is the horizontal position between the pole and furrow. In wild type cells, the cortical tension at the pole and furrow are 1 and 1.8 nN/µm, respectively [10]. In these simulations, we interpolate between these two values according to the measured myosin II concentration (described below). This profile is used as a means of marking intracellular changes in the material properties of the cell during division, not necessarily implying that surface tension comes from myosin. We considered other schemes for spatially varying the cortical tension, but all gave similar results. For example, simulations of cells lacking myosin contractility were run varying cortical tension using a Gaussian distribution. Additionally, we performed simulations using both the myosin density profile and a normal distribution to simulate the surface tension profile but found little difference between the two.

Volume conservation

We assume that the cellular volume remains constant [14]. To enforce this constraint we implement a stress(10)where n is the outward normal. The cell's volume is evaluated by assuming the cell is radially symmetric: V_actual = ∫_{cell length}πr(z) dz. Large values of K_vol keep the cell volume relatively constant, but can lead to small oscillations as the stress overshoots the required target. In our simulations, we set K_vol = 0.1 nN/µm⁵, which was sufficiently high to ensure that both volume changes were small but maintained the stability of the simulations, though some oscillations (as seen in the furrow measurements in Fig. 3E) do appear.

Strain stiffening

We assume that the elastic component of the cell undergoes strain stiffening. Though no precise model for strain stiffening is currently available, in Dictyostelium cells, we have previously observed the effect of nonlinearities in cellular responses to deformations of varying size. These differences depend on the presence of myosin II, likely due to stalling of the myosin II motors [12], [30]. Hence, we posit a plausible phenomenological model of strain stiffening that includes the effect of both the strain (by incorporating the change in the furrow diameter) and the local myosin II-density. The increased elasticity at point x is given by(11)where K₀ is the nominal elasticity (Table 3), myo(z,r) is the myosin density profile (described above) and w_f(t) is the furrow diameter. The resulting temporally and spatially varying map of elasticity is then applied to the material model.

Implementation

The simulations are based on the Level Set Toolbox [40] and are coded in Matlab (Mathworks, Natick, MA). The code is extended to implement the local level set algorithm [41], a modification of the level set method that decreases the computational complexity by solving quantities only near the boundary. Simulations were implemented on a dynamic grid of fixed height (12 µm) and varying width (12–24 µm), with density of 20 points/µm and 5-ms time steps. Simulation takes approximately 2 hours for every minute of cell division on a desk top PC.

Furrow-thinning dynamics

Strains used to determine experimental furrow thinning trajectories are the myoII null (mhcA (HS1):: pLD1A15SN; pDRH:GFP-tubulin) and the rescued myoII null as WT (mhcA (HS1):: pBIG:GFP-myosin II; pDRH:GFP-tubulin) [8], [12]. Time-lapse DIC images of were taken at 2-s intervals with a 40× (N.A. 1.3) objective with 1.6× optivar [8], [12]. To determine the relative furrow diameter, we find the furrow diameter (w_f(t), the diameter of the cell at the midpoint along the z-axis) and the furrow length (L_f(t), the distance between the points of inflection in the furrow region). The point when the two are equal is the cross-over time, t_x and this marks the cross-over distance (D_x = w_f(t_x) = L_f(t_x)). We define the relative furrow diameter as the ratio w_f(t_x)/D_x. Rescaled time is defined by shifting time so that t_x = 0. Furrow diameter and length at each time point were measured using ImageJ (http://rsbweb.nih.gov/ij/).