Modeling the Mechanics of Cell Division: Influence of Spontaneous Membrane Curvature, Surface Tension, and Osmotic Pressure

Abstract

Many cell division processes have been conserved throughout evolution and are being revealed by studies on model organisms such as bacteria, yeasts, and protozoa. Cellular membrane constriction is one of these processes, observed almost universally during cell division. It happens similarly in all organisms through a mechanical pathway synchronized with the sequence of cytokinetic events in the cell interior. Arguably, such a mechanical process is mastered by the coordinated action of a constriction machinery fueled by biochemical energy in conjunction with the passive mechanics of the cellular membrane. Independently of the details of the constriction engine, the membrane component responds against deformation by minimizing the elastic energy at every constriction state following a pathway still unknown. In this paper, we address a theoretical study of the mechanics of membrane constriction in a simplified model that describes a homogeneous membrane vesicle in the regime where mechanical work due to osmotic pressure, surface tension, and bending energy are comparable. We develop a general method to find approximate analytical expressions for the main descriptors of a symmetrically constricted vesicle. Analytical solutions are obtained by combining a perturbative expansion for small deformations with a variational approach that was previously demonstrated valid at the reference state of an initially spherical vesicle at isotonic conditions. The analytic approximate results are compared with the exact solution obtained from numerical computations, getting a good agreement for all the computed quantities (energy, area, volume, constriction force). We analyze the effects of the spontaneous curvature, the surface tension and the osmotic pressure in these quantities, focusing especially on the constriction force. The more favorable conditions for vesicle constriction are determined, obtaining that smaller constriction forces are required for positive spontaneous curvatures, low or negative membrane tension and hypertonic media. Conditions for spontaneous constriction at a given constriction force are also determined. The implications of these results for biological cell division are discussed. This work contributes to a better quantitative understanding of the mechanical pathway of cellular division, and could assist the design of artificial divisomes in vesicle-based self-actuated microsystems obtained from synthetic biology approaches.

Keywords: analytical models; bending energy; cell division; membrane constriction; osmotic pressure; perturbative methods; spontaneous curvature; surface tension.

Figure 1

Sketch depiction of the different modes of deformation possible in a flexible membrane under the action of a constriction force (F_c) (representative of the constriction deformations in cellular plasma membranes at the site of cell division), and of external stress fields applied either transversally, as a hydrostatic osmotic pressure (Δp), or longitudinally, as a lateral membrane tension (Σ). Positive osmotic pressure (Δp > 0), represents a cell at the inflated state of turgor, whereas negative osmotic pressure (Δp < 0) is identified with a flaccid cellular membrane in a hypertonic medium. Regarding lateral membrane tensions, positive surface tension (Σ > 0) represents biological situations of membrane tension underlateral extensional stresses induced by cortical tensions induced by either the eukaryote cytoskeleton, or the peptidoglycan layer in bacteria; negative surface tension (Σ < 0) represents situations of regulated creation of membrane area under in situ membrane biogenesis, or membrane uptake from membrane shuttles coming from the metabolic route of lipid synthesis (lipid trafficking).

Figure 2

Cartoon illustrating how local membrane curvature is determined by the molecular structure of the constituting lipids. Usual phospholipids with a cylindrical molecular aspect assemble as planar membrane aggregates (only a monolayer is shown). In this case (central panel), the equilibrium configuration essentially corresponds to flat bilayer with a zero spontaneous curvature. Charged phospholipids, or lysed species with only one acyl chain present, which show an inverted-cone molecular aspect, cause the membrane to spontaneously bend in a convex configuration (left panel). Inclusion of these membrane molecular formers with a bigger polar head than the thin hydrophobic counterpart leads to situations with positive values of local spontaneous curvature (C₀ > 0). Conversely, cone-like phospholipids (right panel), with a big hydrophobic counterpart thicker than the polar head, leads to membrane aggregates with a concave configuration, which represents an equilibrium bending characterized by a negative spontaneous curvature (C₀ < 0).

Figure 3

(A) Profile R(x) of a symmetrically constricted vesicle with the axis of symmetry along the x–axis and its characteristic parameters. Left polar cap is shaded in yellow and the left half of the constriction zone is shaded in blue. (B) Surface obtained from the revolution around the x–axis of the previous profile R(x).

Figure 4

Shapes at various constriction stages characterized by the constriction parameter s (Equation 10) (s = 0; s = 0.25; s = 0.5; s = 0.75; and s = 0.9) for initially unconstricted prolate, spherical, and oblate vesicles. Prolate case (L_p = 1.5R_m) with Λ = 0.5, spherical case (L_p = R_m) with Λ = 1, and oblate case (L_p = 0.5R_m) with Λ = 1.5, and Γ = −24 for the three cases represented (see Equations 24 and 25) for the definitions of Λ and Γ in terms of C₀R_m, $\tilde{\Sigma }{R}_m^2$ and ($\Delta \tilde{p}{R}_m^3$). When Γ > −29: Λ < 1 gives prolate polar caps (i.e., L_p > R_m), Λ = 1 gives spherical polar caps (i.e., L_p = R_m), and Λ > 1 gives oblate polar caps (i.e., L_p<R_m); while when Γ < −29 the opposite relation between the values of Λ and the shape of the polar holds.

Figure 5

More relevant properties of a constricted vesicle with zero spontaneous curvature C₀ = 0 at all stages of constriction for different values of the dilatation invariant products $\tilde{\Sigma }$${\text{R}}_{\text{m}}^2$ and $\Delta \tilde{\text{p}}{\text{R}}_{\text{m}}^3$, which are associated with surface tension and pressure, respectively. Total energy E_T in units of 8πκ (A), constriction force F_c in units of R_m/κ (B), total area A in units of $4\pi R_m^2$ (C), total volume V in units of $4/3\pi R_m^3$ (D), constriction length L_m in units of R_m (E) and polar distance L_p in units of R_m (F). Comparison between the exact numerical results (points) with the approximate analytical expressions obtained up to sixth order of perturbation (lines).

Figure 6

More relevant properties of a constricted vesicle at all stages of constriction for Σ = Δp = 0 and three values of the product C₀R_m, corresponding to have negative, zero, and positive spontaneous curvature. Total energy E_T in units of 8πκ (A), constriction force F_c in units of R_m/κ (B), total area A in units of $4\pi R_m^2$ (C), total volume V in units of $4/3\pi R_m^3$ (D), constriction length L_m in units of R_m (E) and polar distance L_p in units of R_m (F). Comparison between the exact numerical results (points) with the approximate analytical expressions obtained up to sixth order of perturbation (lines).

Figure 7

Constriction force F_c compared with the reference constriction force F_{c, 0} (defined as the constriction force in the case with C₀ = Σ = Δp = 0), both forces are computed at the beginning of constriction (s = 0.2). ( F_c > F_{c, 0}, F_c < F_{c, 0}, F_c imaginary: impossible constriction). Constriction force is shown as a function of $\tilde{\Sigma }{R}_m^2$ (y-axis) and $\Delta \tilde{p}{R}_m^3$ (x-axis) with C₀R_m = −0.3 (A), C₀R_m = 0 (B), and C₀R_m = 0.3 (C). Regions shaded in blue (red) correspond to conditions giving constriction forces lower (larger) than F_{c, 0} and regions shaded in black correspond to conditions under which constriction is impossible (imaginary analytical results and no numerical solution).

Figure 8

Constriction force at the beginning of constriction (s = 0.2) as a function of $\tilde{\Sigma }{R}_m^2$(y-axis) and $\Delta \tilde{\text{p}}{R}_m^3$ (x-axis) with C₀R_m = 0.8 (A) and with C₀R_m = 1.5 (B). ( F_c > 0: Non-spontaneous constriction, F_c < 0: Spontaneous constriction, F_c imaginary: impossible constriction). The regions shaded in green correspond to the cases in which an external force is required for constriction. The region shaded in orange corresponds to conditions leading to spontaneous constriction (negative constriction forces). Finally, the regions shaded in black correspond to conditions under which constriction is impossible (imaginary analytical results and no numerical solution. If C₀R_m > 1, it is possible to get spontaneous constriction for a certain range of the products $\tilde{\Sigma }{R}_m^2$ and $\Delta \tilde{p}{R}_m^3$, but if C₀R_m < 1 there is no combination of surface tension and osmotic pressure leading to spontaneous constriction.