As previously stated, the natural cell is depicted as a vesicle compartment enclosed by a lipid bilayer membrane in which a given constriction force is applied to create a circumferential furrow positioned at the cell equator. The cellular membrane is characterized by bending rigidity, spontaneous curvature and surface tension. The turgor of the vesicle is maintained under a positive difference of osmotic pressure between the inside cell and the outside extracellular milieu, which represents hypotonic conditions. Flaccid configurations are defined, in general by iso-, hypertonic conditions characterized by zero, or negative, osmotic pressure. Whereas positive membrane tension represents tensioned membrane vesicles forced to create area at the expenses of delivering work of dilation, negative membrane tension will be allowed to consider flaccid vesicles under continuous membrane biogenesis. The problem will be considered in the regime where mechanical work due to osmotic pressure, surface tension, and bending energy are comparable. We extend here the technique presented in Almendro-Vedia et al. (2015), which combines a perturbation expansion for small deformations with a variational approach to compute the minimum energy shapes during the symmetrical constriction of a tensionless vesicle. In such reference problem, a flaccid vesicle was assumed to be constricted at isotonic conditions, and to have a homogeneous membrane with zero spontaneous curvature and negligible tension. In that case, the initial configuration was a spherical vesicle.

The constriction region was described with approximate solutions based on trigonometric functions, whose local curvature is allowed to change depending on the constriction stage. We found previously in Almendro-Vedia et al. (2013) that such ansatz accurately reproduced the results of numerical computations in a broad range of constriction stages. When the spontaneous curvature, surface tension, and osmotic pressure are non-zero, the initial equilibrium configuration of the vesicle (or the cell) is, in general, non-spherical, but can be approximately represented by an ellipsoid, which can be oblate, prolate or spherical in function of the specific values of these parameters. This represents a more general physical scenario and lets us analyze the effect of the spontaneous curvature, the surface tension and the osmotic pressure on vesicle constriction and explain in more detail the biological and physical meaning of these quantities. Spontaneous curvature, C₀, describes membranes with possible asymmetries in the two lipid monolayers resulting in a convex (C₀ > 0), flat (C₀ = 0), or concave (C₀ < 0) membrane at mechanical equilibrium (see Figure 2). The symmetrical case results in a flat membrane at mechanical equilibrium (C₀ = 0), which was the case discussed in Almendro-Vedia et al. (2015). Surface tension, Σ, is defined as the mechanical work per unit area required to increase the membrane area (Σ > 0). It allows to describe different extensional states of biological membranes as the key regulator of cell surface mechanics (Booth et al., 2007; Lecuit and Lenne, 2007). The differential pressure between inside and outside the vesicle, Δp, usually realized as an osmotic pressure, gives the work per unit volume to increase the vesicle volume and allows describing different turgor states of the constricted vesicles, or cells, either turgid (Δp > 0), or flaccid (Δp ≤ 0). Specifically, we consider analytic solutions to the general problem of a constricted vesicle constrained by non-zero values of spontaneous curvature, osmotic pressure, and membrane tension, in the regime where these effects and bending energy are comparable, this is when C₀R_m ≈ 1, and ΔpRm2≈ΣRm2 ≈ κ(1-C0Rm)2 (R_m being the vesicle radius, and κ the bending modulus).

The membrane of a vesicle, or of a living cell, is composed of a lipid bilayer with a thickness that is much smaller than the dimensions of the vesicle. Therefore, the lipid bilayer can be represented approximately by a two-dimensional mathematical surface in the context of the mechanics of the whole cell. In 1973, Helfrich (1973), proposed a simple expression for the bending energy of a membrane in terms of the contributions from mean curvature H (first term, E_m) and Gaussian curvature K (second term, E_G), which are the two geometrical invariants that define the local curvature of the membrane:

Here, Ω is the closed surface that defines the membrane vesicle, and dA its element of area. The parameters κ and κ_G are the bending modulus and the Gaussian bending rigidity, respectively. The spontaneous curvature, C₀, permits to describe bilayers that are spontaneously curved in their equilibrium state due to the compositional inhomogeneity between the inner and the outer monolayers. This term represents the spontaneous tendency of the membrane to build up in a concave (as C₀ < 0), convex (as C₀ > 0), or flat (as C₀ = 0) surface (see Figure 2). In this work, we assume that C₀ is uniform over the vesicle.

In terms of the local principal curvatures of the membrane surface, C₁ and C₂, we have H = (C₁ + C₂)/2 and K = C₁C₂, and the bending energy of the vesicle takes the form:

For a spherical shell of radius R₀, C₁ = C₂ = 1/R₀, the bending energies are Em(sph) = 8πκ(1-R0C0/2)2 and EG(sph) = 4πκG for the mean and Gaussian contributions, respectively. A non-zero value for the spontaneous curvature has strong effects on the configuration of the spherical shell. First, it introduces a characteristic length scale lc≈ C0-1, differently to the case of zero spontaneous curvature for which the deformation energy is a size invariant, this is E_b(C₀ = 0) = 8πκ + 4πκ_G. Since the bending energy of a spherical shell with C₀ ≠ 0 is dependent on R₀ as Eb(R0;C0) = 8πκ(1-R0C0/2)2 + 4πκG, it minimizes at a radius R0=Rmin = 2C0-1, with the evident consequence that the spherical shell with the lowest bending energy Eb(min)(Rmin) = 4πκG corresponds to the particular size Rmin=2C0-1 at C₀ ≠ 0. This conclusion is true for arbitrary shapes (Boal, 2012), meaning that the bending energy is a function not only of cell shape but also of cell size at C₀ ≠ 0. In addition, the sign of C₀ influences the favored shape of the deformed vesicle (Boal, 2012); predominantly convex pear-like shapes are preferred if C₀ > 0, and predominantly concave shapes are favored if C₀ < 0 (Figure 2).

Under the osmotic pressure Δp (the inner minus the outer pressure; Δp > 0 for an inflated vesicle) and under the action of the surface tension Σ, the total energy of a vesicle is given by:

where dV is the element of volume enclosed by the vesicle. Here, we assume that Σ is uniform along the membrane surface and Δp isotropic. Consequently, we can express the Equation (3) as:

Obviously, changing the membrane shape from its equilibrium configuration changes the total energy (Equation 4). However, the Gauss-Bonnet theorem¹, shows that the integral over the Gaussian curvature, the second term in Equations (1) and (2), is constant for surfaces with the same topology. Since the constriction process in a vesicle does not change its topology, and only involves shapes that are topologically equivalent to a sphere (no holes), the contribution of E_G can be ignored because it remains constant, with EG = EG(sph)=4πκG, independently of the size and shape of the vesicle. For the final state, in which the vesicle splits into two separated daughters, it is required to consider the Gaussian contribution since the topological change to two spheres requires an increase of curvature energy by ΔE_G = 4πκ_G. Therefore, during the constriction process (before the final fission), we only analyze the variations of energy due to the mean curvature E_m and the effects of osmotic pressure and surface tension.

We consider the particular case of axisymmetric shapes with the axis of symmetry along the x–axis. When these shapes are represented in Cartesian coordinates as r→=(x,y,h(x,y)) with h(x, y), the surface profile can be given as a height on the x − y plane:

where R(x) is the functional form describing the membrane profile in the x − z plane (see Figure 3A). If the membrane surface is located between x_i and x_f, its bending energy is given by Boal (2012):

with the kernel

where R_x = ∂R/∂x and R_xx = ∂R_x/∂x are, respectively, the first and second derivatives of the membrane profile R(x). Furthermore, other relevant vesicle properties can also be computed, particularly the membrane area and the volume enclosed. For a given profile shape R(x), the area of the corresponding revolution surface around the x–axis is:

and the volume enclosed by this surface is:

Note that Equations (7)–(9) are independent of the coordinate y, as expected for surfaces with rotational symmetry around x. Along the constriction pathway, the vesicle will take the shapes that minimize the total energy E_T (up to thermal effects). In particular, E_T must be stationary under an infinitesimal scale transformation r→→λr→ with small λ − 1. This leads to the following transformations (Seifert and Lipowsky, 1995), κ → κ, C₀ → C₀/λ, C₁ → C₁/λ, C₂ → C₂/λ, A → λ²A, V → λ³V, Σ → Σ/λ², and Δp → Δp/λ³. This means that the shape that minimizes the energy with C₀, Σ, and Δp, has the same energy (and also minimize the energy) under an overall dilatation r→→λr→ with C₀ → C₀/λ, Σ → Σ/λ², and Δp → Δp/λ³. Note that when C₀ = Σ = Δp = 0, the total energy of the vesicle, which is equal to the bending energy, becomes size invariant. This particular case was previously studied by us (Almendro-Vedia et al., 2013, 2015). Here, we consider the more general case, where C₀, Σ, and Δp are non-zero, and analyze the effects of these parameters for the more relevant properties of the system.

We consider the constriction process of a membrane vesicle with rotational symmetry around the longitudinal axis and with central symmetry. The break of the central symmetry can be treated as a stability problem against a linear perturbation from the symmetric case (Almendro-Vedia et al., 2013, 2015). The initial vesicle deforms by the action of a radial tension exerted as a constriction ring at its equator, which decreases the equator radius till formation of a saddled neck that becomes thinner and thinner under the action of the constriction force (see Figure 3). These processes will be followed by the vesicle splitting into two separated daughter vesicles. In previous papers (Almendro-Vedia et al., 2013, 2015), we restricted the study to the case of zero spontaneous curvature C₀ = 0, negligible tension Σ = 0, and no pressure difference between internal and external media Δp = 0, in which the total energy of the vesicle corresponds exclusively to the bending energy (up to thermal effects). In that particular case, the unconstricted initial configuration is a sphere of radius R_m and the constriction process is assumed to proceed by keeping this maximum radius R_m constant, which is equivalent to consider that the two polar caps are hemispheres of radius R_m during the whole process. In the present case, as the parameters C₀, Σ, and Δp are non-zero, the initial configuration is not, in general, a sphere of radius R_m, but a spheroid with polar radius R_m (distance from the center to the upper pole of the spheroid). This spheroid can be an oblate spheroid (when the polar distance L_p, see Figure 3A, is smaller than R_m), a prolate spheroid (when the polar distance L_p is greater than R_m) or a sphere (when the polar distance L_p is equal to R_m). The value of the dimensionless ratio L_p/R_m will depend on the particular values of the dilatation invariant products C₀R_m, ΣRm2, and ΔpRm3. As in Almendro-Vedia et al. (2013, 2015) the constriction is assumed to proceed by keeping the polar radius R_m constant, which implies that, by fixing constant C₀, Σ, and Δp, the shape of the polar caps remains equal to the initial configuration at all stages of constriction. Consequently, the total energy of the polar caps does not change during constriction, making all energy variations arise from central constriction region that goes from R_c = R_m to R_c = 0 (see Figure 3). The case of constant R_m may describe cells whose structure or contents (cytoskeleton, peptidoglycan wall, nucleoid exclusion) exert an effective line tension at the maximum radius sites toward the exterior, Σ_m. Similarly, the force needed to constrain up to a radius R_c is delivered by an effective line tension Σ_c around all the constriction ring and directed toward the cell interior (see Figure 3A). Each constriction state can be characterized by a constriction parameter s, which is defined in terms of the ratio between the constriction radius R_c and the polar radius R_m in the form:

This parameter increases from s = 0 when there is no constriction and R_c = R_m to s = 1 when the constriction is maximal and R_c = 0 (see Figure 4).

Once the origin of the x coordinate is established in the middle point of the vesicle (see Figure 3), the constriction profile is given by:

where L_m and L_p represent the half of the length of the constriction zone and the polar distance, respectively (see Figure 3A). Note that the constriction profile must be continuous in the boundaries of the zones. In addition, since the shape has central symmetry we have the relation R_{left pole cap}(x) = R_{right pole cap}(−x).

The perturbative method allows us to easily obtain approximate analytical formulas in terms of C₀, Σ, Δp and the scaling parameter R_m. We need to determine the shapes that minimize the total energy of the vesicle along the constriction pathway, i.e., for each constriction stage s between 0 and 1. In order to determine the approximate shape we assume an appropriate ansatz for R(x) in each of the existence intervals. These ansatzs will be expressed in terms of the characteristic length rates of each zone: L_p/R_m for the polar caps and L_m/R_m and s for the constriction zone (recall that the polar caps remain constant independently of s). The constriction profile R(x), together with its first and second derivatives, R_x = ∂R/∂x and R_xx = ∂R_x/∂x, allow us computing the integrand K_m (Equation 7) in each zone. Recall that we are assuming that C₀ and Σ are uniform along the whole surface. In order to apply the perturbative method it is convenient to define a small deformation function and expand K_m in power series of it and of its first- and second-order derivatives. Then, introducing this simplified integrand in the total energy (Equation 6), we can perform the integration between the two boundaries that define the corresponding surface interval (Equation 11). Finally, the total energy minimization with respect to the characteristic length of each zone provides their optimal values:

As the polar caps do not change their shape during the whole constriction process, the polar distance L_p (Equation 12a) is independent of the constriction parameter s, (and therefore, the other properties of the system calculated on the polar caps zone are independent of s too). However, the length of the constriction zone changes with the constriction parameter s. Once the optimal total length Lpopt+Lmopt(s) is obtained, it is possible to determinate approximate analytical expressions for the more relevant properties of the system as are the total energy, the membrane area, the volume enclosed and the constriction force at any stage of constriction. These quantities will have the form of a series expansion in powers of the constriction parameter, s. In general, we found that, as expected, that the higher the order included, the better the predictions obtained.

The polar caps of a tensionless vesicle (Σ = 0), without pressure difference (Δp = 0) and for zero spontaneous curvature (C₀ = 0) with maximum radius R_m fixed constant, remain as hemispheres of radius R_m during the whole constriction process (Almendro-Vedia et al., 2013, 2015). However, if these parameters (Σ, Δp, and C₀) are not zero, we have to consider a more general profile for the polar caps. We consider here an ellipsoid with semi-axis R_m (polar radius) and L_p (polar distance) centered in x = L_m (see Figure 3A) with rotational symmetry around x-axis given by:

with x∈ [L_m, L_m+L_p]. Now, in order to apply the perturbative method, we define the small deformation function in the polar caps zone as (see Figure 3):

which leads to:

The global sign of ε determines the shape of the polar caps. When ε is negative, positive, or zero, the polar caps are oblate, prolate, or spherical, respectively.

Once we have calculated the length L_p, other relevant magnitudes can be obtained, particularly the membrane are of the polar caps and the volume enclosed on them. Using the expressions for a surface of revolution (Equations 8 and 9) with the profile of the Equation (13), and integrating between the limits x_i = L_m and x_f = L_m + L_p (see Figure 3A) we obtain for the membrane area:

where Asph = 4πRm2, and for the volume enclosed:

where Vsph=4/3πRm3. Note that the (Equation 16) can be also obtained by expanding the surface area of the spheroid around ε = 0, Aspheroid = 2πRm2 [1 + Lp/(Rme)ArcSin(e)] with e2 = 1-(Rm/Lp)2 = ε(ε + 2)/(1 + ε)2; and the (Equation 17) is exact and corresponds to the volume of the spheroid, Vspheroid = 4/3πRm2Lp.

In Almendro-Vedia et al. (2015), we used the variational approach to find the shape that minimize the energy for different constriction stages in the case of C₀ = 0, Σ = 0 and Δp = 0. There, we considered a family of solutions of the form:

in order to describe the constriction region, where the assumed zeroth-order function family was:

This simple zeroth-order provided good approximations for low and intermediate constriction regimes, as we previously saw in Almendro-Vedia et al. (2013). Consequently, we use here this term as the profile of the constriction region in order to apply the perturbative method to the general case. We define the small deformation function in the constriction region as:

Introducing R_cz(x; s) in terms of u(x; s) in the kernel of the total energy (Equation 7) and expanding it up to the fourth-order of perturbation in u (a higher-order expression can be found in Section 2 of SM) we obtain:

As in the polar caps zone, once we know the dimensionless ratio L_m/R_m, we can determine other relevant vesicle properties, as are the membrane area of the constriction zone and the volume enclosed on it. Introducing the functional form R(x) in terms of the small-u(x) variable (Equation 20) in the formula of the membrane area (Equation 8) and expanding the integrand in a Taylor series up to fourth order in u, we obtain:

Similarly, expressing the integrand of the formula of the volume enclosed (Equation 9) in terms of the small variable u(x) (Equation 20) we obtain the exact result:

Analytical formulas derived with the perturbative method are compared with the (exact) solution of the Euler-Lagrange equations computed numerically. The Euler-Lagrange equations do not have an analytical solution in general, but can be solved numerically and different methods have been developed to solve them. As we previously made when we studied the case with C₀ = Σ = Δp = 0 in Almendro-Vedia et al. (2015), we use the methodology proposed in Jülicher and Lipowsky (1996) and Seifert and Lipowsky (1995), and apply it to axisymmetric shapes subject to equatorial constriction stress with polar radius R_m maintained constant (see Section 1 of SM for a brief explanation of the numerical procedure followed).

Experimental measurements of the bending modulus κ of lipid bilayers in the fluid state give values in the order of 10⁻¹⁹J, or 10−20 k_BT at ambient temperatures (Marsh, 2006; Rodríguez-García et al., 2009; Boal, 2012; Nagle, 2013). They are obtained mostly either from analysis of thermally induced bending fluctuations, or more recently from pipette-aspiration techniques (Marsh, 2006). Observations of the phase behavior of lipid bilayers suggests that κ_G ≈ −0.8κ (Siegel and Kozlov, 2004; Marsh, 2006), which yields a Gaussian energy contribution to the energy of membrane fusion 4πκ_G in the order of 10⁻¹⁸J, or 100k_BT at ambient temperatures (Mingyang et al., 2012). Recall that the Gaussian curvature energy is constant for surfaces with the same topology, independently of the size and shape of the surface. Only when fusion process happen, will the Gaussian energy contribution be considered.

In this subsection we show the approximate analytical expressions obtained for both polar caps and constriction zone in the regime where mechanical work due to osmotic pressure, surface tension, and bending energy are comparable, i.e. Δp Rm3  ≈ ΣRm2 ≈ κ(1−C0Rm)2. A similar condition applies for the spontaneous curvature, which is restricted to the interval −1 ≤ C₀R_m ≤ 1. We have derived the expressions of the characteristic length, the total energy, the membrane area, and the volume enclosed in both zones. These expressions are given in terms of the spontaneous curvature C₀, the surface tension Σ, the osmotic pressure Δp, and the maximum radius R_m. In the constriction zone these expressions also depend on the constriction parameter s (recall that the polar caps remain constant independently of s). Finally, we have obtained the constriction force from the variations in the total energy during the constriction stage.

In this subsection, we analyze the effects of osmotic pressure Δp and surface tension Σ in the more relevant properties of a membrane vesicle with zero spontaneous curvature. The lower values of surface tension and the osmotic pressure difference, the lower the energies of the vesicle and the smaller constriction forces required (see Figures 5A,B, respectively). This means that membranes with small or negative tension (Σ ≤ 0) and immersed in a hypertonic medium (Δp < 0) have less energy and constrict more easily than tensioned membranes (Σ > 0), immersed in an isotonic or hypotonic medium (Δp ≥ 0). As we noted in Almendro-Vedia et al. (2013, 2015), a kick-off force is required to initiate constriction from the initial configuration. However, once the symmetry is initially broken, smaller forces are sufficient to advance cell constriction. At the high constriction regime, the constriction force increases in order to overcome the curvature barrier involved in the pre-fissioned state (see Figure 5B). The total energy of the system (see Figure 5A) increases along the constriction pathway up to double at maximal constriction (when s → 1). In the final two-spheres fission state we have to consider the additional Gaussian curvature energy contribution of 4πκ_G ≈ −100k_BT, since κ_G ≈ −0.8κ (Siegel and Kozlov, 2004; Marsh, 2006), in order to account for the topological change occurred. Inflated vesicles immersed in a hypotonic medium (Δp > 0) have more volume than vesicles immersed in an isotonic medium (Δp = 0), and these last have more volume than shrunk vesicles immersed in hypertonic medium (Δp < 0), (see Figure 5D). This inflation-shrinking process is explained by the osmotic turgor of the living cells. When vesicles (or cells) are placed in a hypotonic medium, water rushes into the membrane increasing the volume of the vesicle. In contrast, when vesicles are placed in a hypertonic solution, water flows out of the vesicle into the surrounding solution, decreasing its volume. When the surface tension is positive, the increase in membrane area is lower under constriction, and vice versa (see Figure 5C). This is explained since positive surface tension implies a tensioned status of the membrane vesicle, which describes biological situations of positive cortical tension with high energetic cost for membrane area extension (Lecuit and Lenne, 2007). Conversely, negative surface tension implies a floppy status, which describes biological situations with a low energetic cost for membrane area extension (Masters et al., 2013). Negative surface tension is equivalent to a net production of membrane, which actually request a negative mechanical work for membrane dilation (Solon et al., 2006). Finally, the lower values of Σ and Δp, the larger the reduced constriction length L_m/R_m (see Figure 5E). Moreover, when Δp and Σ are positive, L_p/R_m < 1, and the polar caps are oblate (see Figure 5F), which corresponds to an inflated vesicle with a tensioned membrane. In contrast, when Δp and Σ are negative, L_p/R_m > 1, and the polar caps are prolate, which corresponds to a deflated vesicle with a tensionless membrane. If Σ = Δp = 0, L_p/R_m = 1, then Λ = 1, and the polar caps are spherical. This result let us relate the shape of the polar caps with the properties of the system. Vesicles with oblate polar caps require more constriction force and contain more energy, less membrane area, and less volume enclosed than vesicles with prolate polar caps. The values Σ˜Rm2 = ±0.3 and Δp˜Rm3 = ±0.3 used in Figure 5 correspond to Σ = ±1.2 × 10⁻⁸ N/m and Δp = ±1.2 × 10⁻² N/m², respectively, a realistic set of values reasonably compatible with a cell-sized artificial vesicle (R_m≈1μm) (Claessensa et al., 2008; Ogleçka et al., 2012) with a relatively flexible lipid bilayer membrane (κ ≈ 10 − 20 k_BT) (Marsh, 2006; Rodríguez-García et al., 2009; Boal, 2012; Nagle, 2013) (Equation 26). Specifically, giant unilamellar vesicles with sizes ranging a few microns, subjected to osmotic stresses of the order of 10 mOsM as much (Δp < 0.01Pa), normally exhibit a lateral tension of the order of 10⁻⁹−10⁻⁸ N/m (Käs and Sackmann, 1991; Rodríguez-García et al., 2009).

In this subsection, we present the results for the constriction process of a membrane vesicle with negligible surface tension (Σ = 0) and no osmotic pressure difference (Δp = 0) for two values of the product C₀R_m, corresponding a positive and a negative spontaneous curvature, respectively. In this way, we can analyze the effects of having a convex (C₀ > 0) or a concave (C₀ < 0) membrane with respect to the flat configuration (C₀ = 0), see Figure 2. Although here we are considering constant spontaneous curvature, recall that, in general, C₀ is not uniform over the membrane of a real cell (Emoto et al., 2005; Renner and Weibel, 2011). The spontaneous curvature has an important effect on the constriction force (see Figure 6B), and a concerted inhomogeneous distribution may play a crucial role in coordinating the contractile rearrangement with the membrane remodeling during cytokinesis (see Discussion). We see that membranes with global positive spontaneous curvatures are more easily constricted (require smaller constriction forces) than flat membranes. In contrast, membranes with global negative spontaneous curvature need higher constriction forces. As in Section Osmotic Pressure and Surface Tension Effects with No Spontaneous Curvature, we can relate the constriction force required with the shape of the polar caps (see Figure 6F). Vesicles with oblate polar caps (when L_p/R_m < 1) require more constriction forces than vesicles with prolate polar caps (when L_p/R_m > 1). The total energy of the vesicle increases as a function of the stage of constriction up to near double its value at the final stage (see Figure 6A). The energy of the vesicles whose membranes have global negative spontaneous curvature is greater than the energy in the flat configuration, while for membranes with global positive spontaneous curvature the energy is lower. As Σ = Δp = 0, the total energy of the vesicle in Figure 6A is exclusively the bending energy. As in Section Osmotic Pressure and Surface Tension Effects with No Spontaneous Curvature, in the final fissioned state we have to consider the additional Gaussian curvature energy contribution of 4πκ_G ≈ −100k_BT (Siegel and Kozlov, 2004; Marsh, 2006), in order to account for the topological change. The increase of the membrane area, vesicle's enclosed volume and constriction length along the constriction pathway is shown in Figures 6C–E, respectively. Vesicles with C₀ < 0, which have oblate polar caps (see Figure 6F), have less area, less volume, and less constriction length than vesicles with C₀ > 0, which have prolate caps. The values C₀R_m = ±0.3 used in Figure 6 correspond to C0 = ± 0.3μm-1 for a cell-sized vesicle (R_m ≈ 1μm).

Finally, we address the analysis of the constriction force when C₀R_m, Σ~Rm2, and Δp~Rm3 are different from zero simultaneously. Figure 7 shows three plots with the different regimes of spontaneous curvature: 7A with C₀R_m = −0.3, 7B with C₀R_m = 0, and 7C with C₀R_m = 0.3, varying Σ~Rm2 between -0.6 and 0.6 (y-axis) and Δp~Rm3 between −0.3 and 0.3 (x-axis). We have calculated the constriction force at the beginning of constriction (s = 0.2) and compare it with the reference constriction force F_{c, 0}, defined as the constriction force at this stage in the case of C₀ = Σ = Δp = 0. For given values of Σ and Δp, the larger the positive spontaneous curvature, the smaller is the constriction force required (see Figure 7B). This means that prolate-shaped elongated shapes whose membranes have an uniform C₀ > 0 are more easily constricted, i.e., those cells tend globally to build up in a convex configuration. Membranes with positive spontaneous curvature tend to form vesicles of smaller radius 2/C₀, which favors the formation of two separated vesicles. Thus, introducing a positive spontaneous curvature extends the region of negative constriction force, i.e., the region where constrictions is an energetically favorable process (see Figure 7).

When the product C₀R_m is greater than 1 (this is, when membranes tend to form vesicles with radius smaller than 2R_m, or cylinder sections with radius smaller than R_m), it is possible to get spontaneous constriction for a certain range of the products Σ~Rm2 and Δp~Rm3 (see Figure 8B, shaded in orange). This means that spontaneous constriction can be induced with appropriate low values of surface tension and osmotic pressure. If the product C₀R_m is smaller than 1 (this is, if membranes tend to form vesicles with radius bigger than 2R_m, or cylinder sections with radius larger than R_m), there is no combination of surface tension and osmotic pressure leading to spontaneous constriction (see Figure 8A).

Instead of considering the polar radius R_m constant, other conditions as constant area or constant volume could be additionally addressed. In these cases, a re-dimensioning strategy can be used, as described in the previous works (Almendro-Vedia et al., 2013, 2015) defining a rescaling parameter λ with the following scaling transformations κ → κ, E_T → E_T, C₀ → C₀/λ, C₁ → C₁/λ, C₂ → C₂/λ, A → λ²A, V → λ³V, Σ → Σ/λ², and Δp → Δp/λ³. This parameter corresponds to:

for constant area condition and to:

for constant volume condition. A(s = 0) and V(s = 0) are, respectively, the membrane area and the volume enclosed by the initial spheroid.

Lipid molecules dispersed in water have the property to self-assemble spontaneously into a bilayer membrane. The lipid bilayer constitutes the main structural ingredient of cell membranes, which endows them with a functional mechanics chiefly determined by its intrinsic elasticity and the curvature properties encoded in the topology of the molecular components (Bretscher, 1973). The resistance of bilayers to area compression and area expansion is much larger than their resistance to bending deformations, while, in the fluid state, there is no resistance to shear deformations. The bending rigidities of usual lipid bilayers in the fluid state take values about 10 − 20 k_BT (Marsh, 2006; Rodríguez-García et al., 2009; Boal, 2012; Nagle, 2013). The combination of the minimization concepts discussed in this paper has allowed the systematic exploration of vesicle energetics under constriction geometry. From our analysis of equatorial constriction (Figure 5B), micron-sized vesicles with a flexible lipid bilayer and a zero spontaneous curvature have constriction forces in the range F_cR_m/κ ≈ 15 − 20. This corresponds to effective forces of the order of pico-newtons, or even lower, as for κ ≈ 10 k_BT and R_m ≥ 1μm, one has F_c ≤ 1pN. Inclusion of non-zero values of spontaneous curvature, osmotic pressure and lateral tension leads to significant changes in the specific quantitative conditions necessary for equatorial constriction (see Figures 6–8). However, no essential change is imposed by these constraints in the qualitative picture, which is almost governed by a monotonic increase of the total energies upon increasing constriction, as in the reference case of no constraints (Σ = Δp = C₀ = 0).

Among the more important biophysical consequences of lipid asymmetries, the subsequent spontaneous curvature of the whole membrane has a crucial impact on the shape transformations of artificial lipid bilayer vesicles (Boal, 2012), which exhibit an extreme sensitivity to induced changes in bilayer asymmetry (Berndl et al., 1990). The simplest description incorporating asymmetry between the two monolayers is given by a non-zero spontaneous curvature C₀, in which case the bending energy becomes dependent of size scale C0-1, thus being minimal for initially curved configurations. As a result, the bending energy loses size invariance and becomes depending on vesicle shape and vesicle size. As a consequence, different axisymmetric shapes of minimal bending energy can be obtained by breaking various symmetries of the sphere. Allowing for asymmetry as well as reflection symmetry, one can obtain prolate and oblate ellipsoids, which are defined by rotating an ellipse about its major and its minor axes, respectively. Negative spontaneous curvature (C₀ < 0) determines polar caps with a predominant oblate shape (Λ > 1); conversely, positive spontaneous curvature (C₀ > 0) determines prolate shapes elongated along the x axis (Λ < 1). Because of the lower energy changes of the prolate shape when subjected to equatorial constriction (see Figures 5B, 6B), the radial forces needed to constrict are significantly smaller in the case of prolate shapes (Λ < 1) than in oblates (Λ > 1). According to Equation (2), the energies of vesicle shapes depend on the value of the spontaneous curvature in addition to membrane tension and osmotic pressure difference, thus, we need to expand our parameter space to three dimensions. The results of minimal energy calculations performed on the basis of Equation (2) subjected to the three parameters (C₀, Δp, Σ) are summarized in Figures 5, 6. Total energies increase with increasing constriction from a value compatible with a single vesicle to a 2-fold value compatible with vesicle fission. The sharper energy changes under initial constriction (s < 0.2) are observed for inflated oblates (Δp ≥ 0, Σ ≥ 0, C₀ < 0 thus Λ > 1), which demand on higher positive constriction forces than in the reference case of a floppy spherical vesicle (Λ = 1). For prolates shapes (Λ < 1), the required constriction forces are, in general, smaller than for oblates. Obviously, prolates shapes are easier contractible and stretchable than oblates, which explains the calculated decrease of the constriction force with increasingly positive spontaneous curvature. At large constriction (s → 1), however, total energies vary almost linearly in all cases, which implies a very similar constriction force range F_cR_m/κ ≈ 20, in the whole space of parameters. For a vesicle (or cell) of micrometer size with a flexible membrane with κ ≈ 10 − 20 k_BT (Marsh, 2006; Rodríguez-García et al., 2009; Boal, 2012; Nagle, 2013), in case of favoring constriction under positive global curvature preferring convex prolates shapes, the constriction forces fall in the range of piconewton, below the value expected for zero spontaneous curvature (see Figure 7B).

A non-zero local spontaneous curvature arises primarily from asymmetry factors in the membrane, particularly differences in the shape and aspect of the component lipids between the two sides of the bilayer. Figure 2 depicts the geometry exhibited by some lipid molecules as the driving force that causes spontaneous membrane curvature (Israelachvilli, 1992; Ritacco et al., 2010; Boal, 2012). Most frequent membrane-formers are cylinder-shaped lipid molecules, which are prone to self-assemble as flat bilayers with a zero spontaneous curvature. Lipids with a polar head group area larger than the cross-sectional area measured at the level of the acyl chains show an inverted-cone shape and tend to curve the membrane positively; in other words, they exhibit positive spontaneous curvature and make the membrane prone to convexity. Prototypical of positive-curvature are lyso-phospholipids, which are intermediates in phospholipidic metabolism, resulted from partial hydrolysis and removing one of the acyl chains of the phospholipids. Due to their inverted-cone molecular aspect, these lyso-phospholipids cannot self-assemble as planar bilayers but form inverse hexagonal phase. When incorporated into bilayers, such non-bilayer forming lipids introduce packing stresses, which, in turn, can affect membrane integrity. Although found only in small amounts in biological cell membranes, lyso-phospholipids have a functional role usually related to cell activation and apoptosis (Munder et al., 1979). Conversely, cone-shaped lipids with a small head cross-section as compared to hydrophobic tails, such as polyunsaturated lipids, diacyl glycerol (DAG), phosphatidylethanolamines (PE) and cardiolipin (CL) exhibit a negative spontaneous curvature (Martens and McMahon, 2008), which make the membrane prone to concavity. During cell division, the process of membrane constriction ends up with a separation of the lipid bilayer of the two daughter cells followed by a fusion of the opposite membranes in a region of high concavity that requires dynamic changes of the lateral distribution and the local composition of membrane lipids. Accumulated experimental evidence points out to the possible mechanical role of negative-curvature lipids during late constriction (Emoto et al., 2005; Litvak et al., 2005; van Meer et al., 2008; Donaldson, 2009; Heberle and Feigenson, 2011; Renner and Weibel, 2011). Those findings demonstrate that the localized production of negative-curvature lipids is required for the proper completion of membrane dynamical process in highly negative-curvature sites, where the local accretion of asymmetric lipids may play a crucial role. Therefore, any extended theoretical model of cell division might consider local negative curvature concentrated in the constriction site of the cell membrane. Localized non-zero spontaneous curvature makes the membrane to be locally prone to a specific curvature, convex (C₀ > 0) or concave (C₀ < 0), depending of the sign of the spontaneous curvature (see Figure 2). Since the constriction site is saddle-shaped, local negative values of the spontaneous curvature may contribute to minimize the local bending energy of the membrane, thus making more realistic further models of cell division, specifically, those accounting for the local accretion of negative curvature lipids in the constriction region.

Binary fission and budding are two scission mechanisms exploited by cells in asexual reproduction pathways. Major difference between binary fission and budding is that in budding there is an asymmetric outgrowth from the parent individual vesicle, or cell, producing a bud, but in binary fission the parent symmetrically splits into two more or less identical offspring. In biological cells, budding is a rather frequent event, because it represents the first step in the production of transport vesicles which shuttle between different compartments of the cell. The simplest approach to understand budding involves a consideration of the lateral and transverse organization of lipids within a membrane, which induces spontaneous curvature followed by morphological change. In an early hypothesis to bud formation, Sheetz and Singer suggested that a local change in the surface area of the two monolayers could lead to negative membrane curvature, inducing the formation of a membrane neck (Sheetz and Singer, 1974). However, advanced models for bud formation in biological cells emphasize a chief role for membrane coating proteins (Schekman and Orci, 1996). In biological constriction processes, the membrane undergoes large mechanical deformations. Although lipids may serve to define the site of bud emergence (Lipowsky, 1992), or determine the onset of divisional constriction (Emoto et al., 2005; Renner and Weibel, 2011), there is almost certainly through the direct action of force exerting proteins (Cao and Wang, 1990; Bi and Lutkenhaus, 1991), or curvature-inducing protein coats (Schekman and Orci, 1996; Bashkirov et al., 2008; Boucrot et al., 2012), that the membrane is able to undergo the large mechanical deformations involved. However, in model vesicles, weak external perturbations suffice to lift the equilibrium constraints of constant area and volume, rendering lipid membranes susceptible of spontaneous budding and fission (Lipowsky, 1991; Miao et al., 1991). For instance, vesicles made of lipids with a weakly negative spontaneous curvature are known to undergo the budding transition at increasing temperature (Berndl et al., 1990; Käs and Sackmann, 1991; Dobereiner et al., 1993), which is equivalent to expanding membrane area. In this budding transition an initially spherical vesicle transforms, via prolate- and pear-shaped intermediates, into two asymmetric spheres, one with a daughter bud, which remains connected to the reduced mother vesicle by a narrow neck (Berndl et al., 1990; Käs and Sackmann, 1991; Dobereiner et al., 1993). In general, osmotic gradients are known to induce bio-reminiscent morphological transformations in giant unillamelar vesicles (Ogleçka et al., 2012; and refs. therein). In particular, to realize budding in protein-free vesicles made of (zero-curvature) single lipids requires large excess area in a flaccid configuration (i.e., hypertonic conditions and/or negative surface tension), which induces a spontaneous constriction process that initiates with the formation of a neck and terminates in the scission of the bud. From our calculation, budding and fission are events that could occur spontaneously under sufficiently low (or negative) surface tension (see Figure 8). Obviously, initially prolates shapes and hypertonic conditions decrease the onset for negative constriction force, thus favoring spontaneous constriction in a homogenous vesicle. In experiments with giant vesicles asymmetric budding is observed largely more frequent than much rarer events of symmetric fission (Berndl et al., 1990; Käs and Sackmann, 1991; Dobereiner et al., 1993), a reasonable fact since symmetry breaking tends to minimize the bending energy of the constricted vesicle (Almendro-Vedia et al., 2015). A further complexity that makes lipid vesicles prone to budding involves the consideration of the lateral and transverse organization of mixtures of lipids within the membrane. Changes in the amount of membrane surface giving rise to excess area, or spontaneous curvature, could occur by transbilayer flip-flop movement of phospholipids, or by lipid phase separation leading to a change in the lipid packing density (the case of heterogeneous membranes will be addressed in the next subsection). Asymmetric budding and symmetric fission in vesicles made by a mixture of lipids has already attracted much theoretical interest (Seifert, 1993; Kohyama et al., 2003; Sens, 2004). The models are all based on the minimization of the bilayer energy, but also vary depending on the interactions among the lipids in multicomponent systems, which make them to separate into phases or not. For monophasic, homogeneous vesicles, the membrane neck involved in the budding transition is produced by the lipid molecules whose local negative curvature is different from the main lipids of the membrane. If the molecules prefer a negatively curved bilayer, they will favor the formation of a bud. For biphasic, heterogeneous vesicles, a line tension exists between the two phases, trying to reduce the interface length, and favoring asymmetric budding and symmetric constriction, eventually leading to spontaneous fission (Lipowsky, 1992). This heterogeneous scenario will be further discussed the next subsection. Our results point out the easy practical availability of the onset of spontaneity for the budding/fission transition. Indeed, negative constriction forces are required for relatively low values of membrane tension, even for inflated vesicles under moderately positive osmotic pressure (see Figure 8). Obviously, the possibility for spontaneous budding/fission enhances in prolates shapes defined by high positive values of the global spontaneous curvature (see Figures 7, 8), a fact already recognized in the early studies of the morphological transitions of membrane vesicles (Berndl et al., 1990; Lipowsky, 1991; Seifert and Lipowsky, 1995). The current study allows for quantitatively determining the specific conditions for spontaneous budding/fission from very accurate analytic formulas, which provide an interesting predictive framework for the design of smart vesicle microsystems endowing the division functionality (Osawa et al., 2008).

Any cellular membrane, even in the simplest organisms like bacteria, actually consists of a complex mixture of structural lipids, proteins, and a small amount of functional glycolipids and glycoproteins involved in membrane signaling and trafficking. In the simplest mechanical depiction, a realistic cellular membrane might be modeled as a composite shell (Sackmann et al., 2002), composed by a heterogeneous lipid bilayer and adjoined cortical protein, or glycoprotein structures, such as the inner cell cortex in eukaryotes, or the outer peptidoglycan cell wall in bacteria. Such cortical structures can be described as a rigid cover somewhat connected to the fluid bilayer. From the mechanical standpoint, those rigid structures not only strengthen the lipid bilayer against the bending deformation but also bears in-plane shear, which is not supported by the fluid lipids. If the membrane skeleton, or the bacterial wall, are roughly considered to be structurally continuous, the composite cellular shell can still be regarded as a 2D continuum medium, mechanically described by the current material constitutive modeling, eventually accounting for lateral heterogeneity, plus an additional elasticity modulus describing in-plane shear rigidity. In addition, subcellular localization of the cytokinetic apparatus and related proteins is a universal feature of any prokaryote or eukaryote cell. However, although some targeting anchors are known in some organisms, the origin of polar and division-site localization remains mysterious for a large fraction of cytokinetic proteins. Ultimately, the molecular components responsible for such symmetry breaking must employ a high degree of self-organization, which could contribute with additional ingredients to the mechanics of division. For instance, curvature-induced stabilization mechanisms, based on the spontaneous curvature of localized membrane components, have been proposed to account for spontaneous lipid targeting to the poles and division site of rod-shaped bacterial cells (Huang et al., 2006). In that model, if one of the membrane components has a large intrinsic curvature, the geometrical constraint of the inner lipid membrane by the more rigid bacterial cell wall naturally leads to lipid phase separation, and the resulting clusters of high-curvature lipids are large enough to spontaneously localize at cell poles and division site (Huang et al., 2006), in agreement with the experimental evidence of localization of the phospholipid cardiolipin to the negatively curved regions of E coli membranes (Renner and Weibel, 2011), and polar targeting of some cytokinetic proteins during bacterial division (Huang et al., 2006). In general, aggregates of lipids, proteins, or lipid-protein complexes may localize in response to cell geometry, introducing additional ingredients of membrane mechanics that might be accounted for. Although all this complexity might be included on an extended, more realistic, model of cell division, the physical problem exceeds the limits of the present work, which can be however considered as a good starting point to obtain approximate solutions that offer a general depiction of the minimal energy mechanical pathway of divisional constriction in different organisms under different geometrical and constitutive conditions of their cellular membranes.

The big question, which remains still to be addressed in a comprehensive way, is to know how much of the division of real cells can be understood in terms of a simplified physical model integrating the passive mechanics of the membrane with the active actuation of a cytokinetic engine. In a minimalistic perspective, the division machinery should have evolved to fulfill, at least, the work requirements of reorganizing the cellular plasma membrane along the cell cycle, especially during cytokinesis. Certainly, the membrane deformations involved along the constriction stage of cell division in biological cells request of an expenditure of mechanical work exerted by a cytokinetic engine, which could however be working under conditions of minimal energy consumption, or perhaps nearing the onset for spontaneous constriction.

Evidence of cell division without the action of a cytokinetic machinery is already accepted to exist in complete kingdoms of archea, and in some species of bacteria including Chlamydia and Planctomycetes among others (Erickson and Osawa, 2010). The bacterial cytokinetic protein FtsZ is not present in all those organisms, usually called Dftsz (devoid FtsZ), which cells are able however to undergo membrane constriction and develop two daughters till mature division without FtsZ. Although Dftsz cells spend longer times in division, the constriction slowing-down does not affect their overall growth rate (Erickson and Osawa, 2010). All Dftsz bacteria that divide without FtsZ apparently use their motile apparatus to pull the two daughters apart. Under division the constriction neck appears very elongated, with the pair of daughters connected by a thin extension, similarly to the predictions of our current model at low internal pressures Δp < 0, implying Λ < 1 (Equation 24) (see Figure 4, first column). Longitudinal traction-mediated cytofission appears to be a competent mechanism for cell division in bacterial cells lacking FtsZ. In those cases, the mechanics of the bending deformations coupled with the osmotic conditions should be essential for the correct understanding of this minimal work-demanding divisional mechanism. Spontaneous constriction via excess membrane area (see Sections Spontaneous Curvature Effects and Budding and Spontaneous Fission) seems to be a mechanism for nearly-spontaneous binary fission in some mutant strains of B. subtilis, especially L-forms that grow without a cell wall and divides without FtsZ (Leaver et al., 2009). Detailed observation of the division process of L-forms of E. coli, Listeria, and B. subtilis reveals a two-step mechanism that exploits the large excess area of those Dftsz cells, initiating the process by a membrane extrusion phase that leads to a long protrusion, which resolves by cleaving into smaller round progenies (Leaver et al., 2009; Erickson and Osawa, 2010). A similar nearly-spontaneous division mechanism was discovered in the eukaryote Dictyostelium when its myosin II gen was knocked out (De Lozanne and Spudich, 1987; Knecht and Loomis, 1987). When those myosin lacking cells (thus no cytokinetic motor is working out) are allowed to adhere to a substrate, adhesion forces restore many features of normal furrow constriction and the cells become able to undergo “illegitimate division” by following the same mechanism of traction-mediated cytofission observed in Dftsz bacteria.

Constant maximum radius condition may give a simplified description of cytokinesis of rod-shaped cells, like E. coli and Bacillus subtilis, in which the constant maximum radius is maintained by an external tension (due to a peptidoglycan wall) and represented in our model by the line tension s_m. Under this condition, our model predicts unchanged poles during all the constriction process, which is consistent with the approximately constant shape of the poles in the rod-shaped cells (Field et al., 1999; Cabeen and Jacobs-Wagner, 2005; Reshes et al., 2008).

In addition, our model can be extended to other cases, as constant volume and constant area conditions (see Section Constant area and Constant Volume Conditions). Constriction at constant volume requires a nearly 30% increase in area (Almendro-Vedia et al., 2013), and may describe divided cells with intense membrane trafficking (Morré, 1975; Nohturfft and Zhang, 2009), which is known to play an important role in cytokinesis, (Albertson et al., 2005; Boucrot and Kirchhausen, 2007). On the other hand, if constriction takes place at constant area, the volume must be reduced in approximately 30% (Almendro-Vedia et al., 2013), which may describe divided cells with low or inhibited membrane trafficking. Thus, in constant area constriction, a greater initial area is required to have the same final volume. Heat shock has been shown to increase the area before division (Kutalik et al., 2005; Niven et al., 2008) and to affect membrane trafficking molecules genes expression, but also other genes as those of signaling molecules (Kim et al., 2011).

Normal cell division in evolved cells involves a mature cytokinetic engine able to exert the constriction forces that cleave the cell in the division site. This is an up-hill process that requires the expenditure of an important amount of mechanical work by a constriction machinery. However, other concomitant, perhaps redundant, constriction mechanisms could be working to favor the membrane constriction phase of cell division. Redundant systems are often exploited indeed by living cells (Edelman and Gally, 2001). As refers the constriction process during cell division, local creation of negative spontaneous curvature (C₀ < 0) might contribute to favor constriction; both, hypertonic stresses (Δp < 0) and biogenic processes of membrane creation (Σ < 0) mediated by lipid trafficking also favor constriction. In addition, biological cells undergoing divisional constriction there used to be subjected to area and volume restrictions during the cytokinetic phase of the cell cycle. Despite of his inherent simplicity, all these ingredients are already accounted for by our physical model, which could be used to determine the energy landscape for whole configurational space of geometrical characteristics and constitutive properties captured for different classes of cells, from bacteria to eukaryotes. The optimal pathways for the mainstream mechanism of membrane constriction can be identified on this mechanical landscape, and predictions about changing external field parameters (osmotic stress, membrane tension, etc.) could be realized and checked in view of the experimental observations. Further complexities arising from heterogeneous membrane composition and additional shear rigidity introduced by skeletal structures might contribute to complete the picture.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.