## 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 coupling^{2} in cells [5]. Numerous prior studies have extensively investigated the impact of electric and magnetic fields on the morphology of cells and fluid vesicles [6–14]. 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.

## 2 Physical and Mathematical Model

*linear*dielectric behavior of the membrane (of relative permittivity $\epsilon r$)

^{3}:

*effectively*incompressible. Therefore, we have

For given $\kappa b$, $E0$, $\epsilon r$, $\kappa t$, $\kappa s$, $t0$, 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.

## 3 Stability Analysis

## 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 $\epsilon $ of approximately 0.97 for the bio-related values as shown in Table 1 and a transmembrane voltage of $V0=\u221270mV$.

To provide a clearer illustration, we have plotted the free energy as a function of the aspect ratio $\epsilon $ in Fig. 2 for different initial electric field values of $E0=\u2212V0/t0$. 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 $\epsilon $ 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 ($\epsilon =1$). 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 $\epsilon =1$ 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.

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 ($\u223c107V/m$) and normal size range of animal cells (between 10 and 100 $\mu m$ [23]).

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 ($\u221280$ to $\u221250mV$) [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

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

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.