Abstract

Biological cells are almost never truly spherical, even in the absence of an obvious cause that disrupts the cell’s symmetry. Using rather simple energy considerations, we show that even though all real biological cells are subject to a completely radial electric field along the cell membrane, the spherical shape is unstable under most practical situations. This simple result appears to have been overlooked in the literature.

1 Introduction

The biological cell is considered to be the smallest unit of life. It is enclosed by lipid membranes that contribute in dictating many cellular processes [1]. While modeling biological cells as perfectly spherical is a common practice for gaining insights into cellular functions, it is important to note that, in nature, cells rarely exhibit a flawless spherical morphology. Instead, cells often adopt various irregular shapes, such as elongated or asymmetrical forms. In this paper, we point out a curious result. It is well known that cell membranes possess a cross-membrane resting potential difference due to actively regulated ion transportation (the typical value of this potential difference varies from 40 to 75 mV) [2]. Thus the membrane exhibits a radial preexisting electric field and there is no ostensible disruption of symmetry. Despite this, our mathematical analysis appears to illustrate the celebrated analog of the so-called Rivlin’s cube [3]: for realistic physical parameters, the spherical shape is unstable under a radial electrical field.

We remark that understanding the complex interactions between mechanical deformation and electric fields in cell membranes is an active area of research. We avoid a detailed review of the literature but refer to a recent review article on electro-mechanical coupling2 in cells [5]. Numerous prior studies have extensively investigated the impact of electric and magnetic fields on the morphology of cells and fluid vesicles [614]. These investigations have demonstrated that when subjected to sufficiently high fields along specific axes, ellipsoids of revolution become the favored configurations. Consequently, the transition of vesicle shapes from oblate to prolate has attracted significant attention within the research community. Various approaches have been explored to achieve this shape transition. For instance, researchers have investigated altering the internal conductivity of spherical vesicles [13], implementing stepwise uniform direct current (DC) electric fields [10], or modifying the frequency of an alternating current (AC) field [11,12], among other methods. These studies contribute to our understanding of the mechanisms behind shape transformations in vesicles and provide valuable insights into the diverse ways such transitions can be achieved.

We may intuitively assume that a radial electrical field across the membrane will not break the symmetry of the initial (assumed) spherical shape of a cell. This assumption is reminiscent of the Rivlin’s cube, a well-known problem in mechanics [3]. In the purely mechanical scenario, Rivlin investigated the deformation of an incompressible isotropic cube, with each of its six faces subjected to a positive tensile force F, resulting from uniformly distributed normal tractions on each face. Due to the symmetry of the cube and the incompressibility of the material, linear elasticity predicts no deformation would occur, regardless of the magnitude of the tensile forces. However, Rivlin challenged this assumption by considering the distinction between the initial and deformed configurations and incorporating constitutive nonlinearities. His analysis revealed that the aforementioned assumption was erroneous. The solutions to this problem not only included the unit cube but also many solutions that lead to non-cuboid shapes. The lack of uniqueness in the solutions prompted a stability analysis, which subsequently demonstrated that the cubic configuration becomes unstable under certain conditions [15], favoring some of the non-cubic configurations.

Drawing inspiration from Rivlin’s cube, we are tempted to question whether the assumption of spherical stability under a radial electric field in biological cells is also incorrect. In a related study, Helfrich demonstrated the ability of a radial pressure difference to deform a spherical vesicle into an ellipsoidal shape [7]. However, it is worth noting that Helfrich’s work, although exploring the idea of symmetry breaking in a spherical shape, did not consider the electrical contribution and relied on an elastic model based on assumptions different from those in our current paper. Specifically, Helfrich assumed that the vesicle’s deformation was driven solely by a curvature elastic energy, resulting from a difference between external and internal pressures while keeping the vesicle’s area constant. Moreover, another study [16] utilized Monte Carlo simulations to investigate vesicle shapes in the absence of volume constraints, as well as under the influence of finite osmotic pressure. The findings of this study supported the notion that prolate or oblate shapes are the most probable configurations for vesicles. In the following, we create a model to understand the influence of a radial electric field on an initially spherical cell (Fig. 1). Our model incorporate the elasticity of the biological membrane, encompassing bending, thickness deformation, and stretching to describe its mechanical behavior. We assume that the biological membrane is effectively incompressible and that the volume of the cell remains constant. The only “loading” is the completely radial electrical field due to the resting potential.

Fig. 1
The conceptual schematic of a biological cell subject to a radial electric field along the cell membrane: initial versus deformed configurations
Fig. 1
The conceptual schematic of a biological cell subject to a radial electric field along the cell membrane: initial versus deformed configurations
Close modal

2 Physical and Mathematical Model

We consider an elastic membrane of relative dielectric permittivity εr separating the cell interior from the outside media. We assume that both the exterior and interior media of the cell are conductive. Given the emphasis on nonlinear deformation, it is crucial to maintain a clear distinction between the reference and deformed configurations. Let MR3 be the 3D membrane body with mid-surface being Ω. In the reference configuration when there is no potential difference, the membrane body M0 is a shell of thickness t0 and inner radius R0, satisfying the condition t0R0. The deformation of the membrane, represented by the mapping y:M0M, describes the deformation from the reference mid-surface Ω0 to the deformed mid-surface Ω. The polarization in the membrane in the deformed configuration, denoted as p:MR3, describes the thermodynamic state of the system. Constitutively, we assume linear dielectric behavior of the membrane (of relative permittivity εr)3:
(1)
where e denotes the spatial electric field, and ε0 is the vacuum electric permittivity. Under the application of a cross-membrane resting potential V0, the total free energy of the system can be identified as
(2)
where Uelast is the elastic energy arising from the deformation of the elastic membrane, and Eelct are the electric contributions to the free energy. To simplify our analysis, we assume that the intracellular and extracellular media can be treated as fluids with negligible elasticity. In this regard, we consider the energy penalty arising from bending, thickness deformation, and stretching as a mean to describe the elastic behavior of the cell membrane:
(3)
where Ω=y(Ω0), κb is the bending modulus, κt is the modulus associated with thickness changes (measured in energy per unit area), and κs is the stretch modulus. In (3), H denotes the mean curvature of the deformed configuration (ellipsoid) whereas H0 corresponds to the mean curvature of the reference configuration (sphere). Similarly, t0 refers to the thickness of the membrane in reference configuration and t describes the thickness of the deformed membrane. Moreover, considering the biological membrane as a fluid membrane, we assume that it is effectively incompressible. Therefore, we have
(4)
Furthermore, we assume that the cell volume remains constant during the deformation, and hence we have
(5)
We need to determine the electrical contribution from the resting potential across the membrane. This electrical contribution to the free energy is identified as [17,18]
(6)
where the electric potential φ:MR is determined by the Maxwell equation:
(7)
By definition, the electric contribution is given by Equation (6) and can be rewritten as
(8)
Equation (8) was derived using the divergence theorem. The following identity is obtained
where the last equality follows from the Maxwell equation (7). Applying the constitutive law p=ε0(εr1)φ, by Eq. (6), we obtain
Moreover, since the radius of the cell is much larger than the membrane thickness R0t, the solution to the Maxwell Equation (7) and Equation (1) is approximately given by
(9)
In the following analysis, we strive for conceptual simplicity and a clear understanding of the physical implications of our mathematical model. To achieve this, we solve the free energy (2) based on an assumption about the deformation of the cell membrane. Specifically, we assume that the change in thickness of the deformed membrane M is uniform (t=t¯=const. on |Ω|) and the overall cell is deformed into a spheroid with semi-axis length a,a,c. The free energy is expressed in terms of (a,c,t¯) as follows:
(10)
where |Ω|=4πa2c/3 is the volume of the spheroid, |Ω|=2πa2(1+(c/aesin1)(e)) for a prolate spheroid with c>a, and e=1a2c2. To incorporate the volume constraint from (5), we simply substitute c by R03/a2 into (10). To account for the constraint in (4), we replace t¯ by t0|Ω0|/|Ω| where |Ω0|=4πR02 is the surface area of a sphere in the reference configuration. To visualize the updated energy (10) in terms of the ratio a/c, we introduce ε=a/c and substitute a by R03ε in order to plot it with respect to ε and observe its behavior around ε=1 (spherical shape). For more detailed calculations, the reader is referred to the following derivations. The bending energy depends on the mean curvature of the deformed and the reference configuration. The deformed configuration is a prolate so, unlike the case of a perfect sphere where the mean curvature H0=2/R0, the mean curvature will depend on the radial angle as shown below
(11)
The surface element of a prolate is expressed as follows:
(12)
The bending energy needs to be integrated carefully taking into account the polar angle. The following is obtained:
(13)
The energies associated with the stretch and the thickness change are derived as follows:
(14)
The electric contribution becomes
(15)

For given κb, E0, εr, κt, κs, t0, and R0 values summarized in Table 1, we can plot the free energy and see the minima around the spherical state. The findings will be discussed in Sec. 4.

Table 1

The numerical values used to generate all the results

QuantitiesValues
κb(Nm)1.234×1019 [19]
κt(N/m)0.142 [20]
κs(N/m)0.106 [19]
ε0(F/m)8.854×1012
E0=V0/t0(V/m)1.4107 [2]
εr3 [21]
t0(nm)5 [22]
R0(μm)5 [23]
QuantitiesValues
κb(Nm)1.234×1019 [19]
κt(N/m)0.142 [20]
κs(N/m)0.106 [19]
ε0(F/m)8.854×1012
E0=V0/t0(V/m)1.4107 [2]
εr3 [21]
t0(nm)5 [22]
R0(μm)5 [23]

3 Stability Analysis

Soft materials and in particular biological membranes are well known for their ability to deform significantly, often leading to surface instabilities. While these instabilities are typically seen as problematic in engineering, they can be useful for applications like patterning and energy harvesting [24]. In this section, we would like to investigate the stability condition of biological cells and examine when the spherical shape loses its stability to become a spheroid. As we showed before, the free energy F can be exclusively expressed in terms of the aspect ratio ε. So the stability condition can be deduced from the following:
(16)
If (d2F[ε]/dε2)|ε=1>0, it means that the spherical shape is stable. However, if (d2F[ε]/dε2)|ε=1<0, it means that the ellipsoid is stable. Doing a change of variable by substituting R0 by 33|Ω0|/4π in (16), we can thermodynamically deduce the critical pressure at ε=1 as follows:
(17)
From (17) and knowing that E0=V0/t0, then Eq. (16) becomes
(18)

4 Results and Discussion

Similar energy contributions were accounted for in our prior work [25]. In the current study, we advanced the previous work by incorporating a bending energy penalty term into the free energy formulation, enhancing our ability to explore equilibrium shapes. This approach allowed us to better investigate the underlying reasons behind the breaking of spherical symmetry. The results reveal that accounting for bending energy leads the equilibrium state to converge toward a prolate shape. Specifically, by solving numerically the free energy represented by (10) to find the equilibrium shape, we observed an aspect ratio ε of approximately 0.97 for the bio-related values as shown in Table 1 and a transmembrane voltage of V0=70mV.

To provide a clearer illustration, we have plotted the free energy as a function of the aspect ratio ε in Fig. 2 for different initial electric field values of E0=V0/t0. In Fig. 2(a), we observe that the free energy exhibits two distinct wells for the typical value of the electric field. One of the wells corresponds to the prolate configuration, where the aspect ratio ε is approximately 0.97, confirming our numerical findings. The other well corresponds to the oblate configuration, with an aspect ratio around 1.03. Notably, we observe that the energy reaches a maximum at the spherical configuration (ε=1). In this particular case, the transition from the ellipsoidal shape to the spherical one incurs a relatively high energy barrier of approximately 100kBT. In Fig. 2(b), we demonstrate that by reducing the initial electric field by one order of magnitude (an unusual value for biological cells), the maximum at ε=1 becomes more flattened, and the energy barrier decreases to less than 0.01 kBT, comparable to thermal fluctuations. In other words, any thermal perturbation may cause the cell to transition back and forth between the two states.

Fig. 2
Transitioning from a single to a double energy well
Fig. 2
Transitioning from a single to a double energy well
Close modal

In Fig. 2(c), as the initial electric field is further reduced to 0.14mV/nm, we observe that the energy profile reduces to a single well, leading us to conclude that there exists an electric field threshold above which the cell tends to adopt a symmetry-breaking configuration more readily, while below this threshold, the spherical configuration becomes more favorable and stable.

In Sec. 3, we perform a brief stability analysis around the spherical configuration to determine its stability condition. We find that the positivity of (16) must be satisfied for the biological cell to maintain its spherical shape. Interestingly, we observe that the bending modulus is the only elastic parameter that affects the stability equation, while the stretch modulus and the modulus associated with thickness change do not play a role in maintaining the spherical shape.

Based on the phase diagram shown in Fig. 3(a), which is obtained from (16), it is evident that the spherical shape is unstable for the typical magnitude of the electric field (107V/m) and normal size range of animal cells (between 10 and 100 μm [23]).

Fig. 3
Phase diagrams showing the stability and instability regions of the spherical configuration of a biological cell
Fig. 3
Phase diagrams showing the stability and instability regions of the spherical configuration of a biological cell
Close modal

To gain further insight into our findings, we derive the critical pressure thermodynamically, as shown in (17). From (18), we plot the phase diagram in Fig. 3(b), illustrating the critical pressure as a function of transmembrane voltage and indicating the stability regions of the spherical shape. It is evident that, for typical values of the transmembrane voltage (80 to 50mV) [2], the spherical shape is unstable. However, as we decrease the transmembrane voltage toward atypical smaller values in absolute terms, we eventually reach a stable spherical state, which is confirmed by both Figs. 3(b) and 2(c). Thus, according to our model, we can conclude that the ellipsoidal shape should be prevalent, and cells are unlikely to maintain a spherical configuration.

While the significance of these findings in biophysics is evident from the preceding discussion, we also note their potential relevance in the design of novel devices capable of transitioning between different configurations by manipulating the applied voltage, such as electric pumps. An interesting future analysis may involve inclusion of magnetic fields and flexoelectricity [5,4,26].

Footnotes

2

We ignore the phenomenon of flexoelectricity, c.f. Ref. [4]. While important, this is not germane to the present discussion.

3

The key nonlinearities that must be accounted for are geometric in nature and not constitutive.

Acknowledgment

Author Sharma gratefully acknowledges support from the Hugh Roy and Lillie Cranz Cullen Distinguished University Chair of the University of Houston.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The authors attest that all data for this study are included in the paper.

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