Mechanics of constriction during cell division: a variational approach

Abstract

During symmetric division cells undergo large constriction deformations at a stable midcell site. Using a variational approach, we investigate the mechanical route for symmetric constriction by computing the bending energy of deformed vesicles with rotational symmetry. Forces required for constriction are explicitly computed at constant area and constant volume, and their values are found to be determined by cell size and bending modulus. For cell-sized vesicles, considering typical bending modulus of [Formula: see text], we calculate constriction forces in the range [Formula: see text]. The instability of symmetrical constriction is shown and quantified with a characteristic coefficient of the order of [Formula: see text], thus evidencing that cells need a robust mechanism to stabilize constriction at midcell.

Figure 1. Symmetrically constricted vesicle.

A. Longitudinal section at and characteristic parameters of a deformed vesicle under symmetrical constriction represented on the optimal shape obtained for using first order approach. B. Surface resulting from the revolution of the optimal shape represented in Fig. 1A. C. Transversal section at a given . The height at a given point is given by . Due to rotational symmetry around axis, all transversal sections are circumferences. We denote its radius by . The height and the radius are related by the Pythagoras' Theorem which leads to Eq. (3).

Figure 2. Values of the variational parameters .

Values of the parameter as a function of constriction parameter for different orders of approximation in the variational approach.

Figure 3. Aspect ratio of the constriction region.

Aspect ratio of the constriction region as a function of constriction parameter for different orders of approximation in the variational approach.

Figure 4. Bending energy.

Bending energy (in units of ) as a function of constriction parameter for different orders of approximation in the variational approach. Profiles maintaining constant at different stages of constriction are also shown.

Figure 5. Rescaling parameter , area and volume.

A. Rescaling parameter as a function of constriction parameter , for different cases: constant maximum radius, constant area, and constant volume. B. Area in units of and volume in units of as a function of constriction parameter for different constraints.

Figure 6. Shapes during constriction process.

Shapes at various constriction stages ( = 0,  = 0.01,  = 0.3,  = 0.6 and  = 0.9) with the condition of constant volume.

Figure 7. Constriction force.

Constriction force (in units of ) as a function of constriction parameter . Due to its trend it is divided in three regimes (I, II and III) with different behaviour.

Figure 8. Asymmetrically constricted vesicle.

A. Symmetric and asymmetric constriction optimal shapes [ vs. ] with with the same volume plotted with the characteristic parameters for defining asymmetrical constrictions. B. Asymmetric surface resulting from the revolution along the axis of the asymmetric in Fig. 8A.

Figure 9. Instability coefficient of symmetrical constriction.

Instability coefficient of symmetrical constriction for constant area and for constant volume (in units of ) vs. constriction parameter for different orders of approximation in the variational approach and calculated numerically.