Electrodeformation of multi-bilayer spherical concentric membranes by AC electric fields

The effects of electromagnetic fields on human health have been a concern for a long time and remain a subject of debate. The debate has intensified lately due to the widespread use of cell phones and other electric devices that function close to the human body. The radio frequencies of such devices range from 3 kHz to 300 GHz [1]. Cells in the human body exposed to these fields will respond in a manner still unknown. In vitro experiments have shown that cells can be oriented and deformed with AC fields [2]. The phenomena of dielectrophoresis and electrorotation are also well known [3]. Even more interesting, it is known that the electrical properties of a cell change when pathological conditions develop [4–6]. If electric fields deform the cell, then it is expected that all organelles are affected. Organelles are often enclosed by a lipid bilayer and the nucleus is even surrounded by a double bilayer. The nucleus is the control center of the cell, that is why we focus our attention on this organelle. Indeed, it has been shown that changes in cell shape results in modifications of nuclear shape and functions [7–9]. Cell shape changes in endothelial cells have been associated with nuclear shape remodeling [10] resulting in regulation of gene expression [11,12]. These changes in the nucleus would depend on its mechanical properties, being perhaps more pronounced on cells with softer nuclei. As it turns out, changes in the mechanical properties of cancer cells have been studied in recent years [13], and it has been shown that cancer cells are more compliant than healthy cells [14,15]. However, it has also been shown that infected and cancer cells can be stiffer [16,17] than healthy cells and in some cases, the increase in stiffness can result in apoptosis [18,19]. The effects of AC electric fields on cells are diverse, depending on the target cell and exposure conditions, which makes comparing the experimental results difficults due to the different parameters used (frequency and strength of the electric field, external conditions, different electrode types, and arrangements, etc.). Nevertheless, there is an agreement on the effects on cells at low intensity, intermediate frequency, electric fields. For example, intermediate electric field frequencies inhibit cancerous growth in vitro [20,21] and induce cell cycle arrest and abnormal mitosis by causing the malformation of microtubules that results in cell destruction [22]. In order to study the deformation of cells by AC electric fields we use an oversimplified model of the cell, hereinafter referred to as cell-like system (CLS). Our CLS consists of two concentric spheres, see fig. 1. Each sphere, which is treated as a lossy dielectric, is a bilayer membrane representing the plasma membrane of the cell (external sphere) and the nuclear membrane (internal sphere). Experimentally, CLS are vesicles with sizes of $10\text{--}100\ \mu \text{m}$, known as giant unilamellar vesicles (GUVs), that provide a very simple model for the interaction of electric fields with the cell membrane [23]. When vesicles are exposed to electric fields an electric potential is generated through the membrane and an effective electric tension is induced. The bending and stretching of the membrane by such tension has been studied theoretically [24–26] and experimentally [27]. The main results of these studies demonstrate that vesicle deformations depend on the mechanical properties of the vesicles, the electrical properties of the surrounding media, and the lipid membrane.

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In this paper we calculate the deformations and transition shapes of the nucleus and plasma membrane in a CLS model as a function of the frequency of the electric field and the electrical properties of the media and membranes.

The bilayer membranes in our CLS model have thicknesses l_m and l_n for the membrane and the nucleus, respectively. The deformations of a membrane due to an AC electric field lead to an increase of the bending energy, adopting prolate and oblate shapes. We consider the deformation to be prolate if the polar radius is greater than the equatorial radius of the spheroid, and oblate if the polar radius is smaller than the equatorial radius. In this work, the z-axis is parallel to the polar radius. To describe the change in elastic energy we use the bending energy introduced by Winterhalter and Helfrich [24]

$$\begin{equation} \Delta F_{be(i)} = \frac{48 \pi}{5} \bigg(1-\frac{M_{sp(i)} R_i}{6}\bigg) k_i \bigg(\frac{s_i}{R_i}\bigg)^2; i = 1,3 \end{equation} \tag{ 1 }$$

where, $R_1=r_m$ and $R_3=r_n$, s_i is the deformation amplitude, k_i is the membrane bending rigidity of the membranes, and M_sp(i) is the spontaneous curvature. The membrane and nuclear radii are defined as $r_{m}=r_{2}+l_{m}/2$ and $r_{n}=r_{4}+l_{n}/2$, respectively, see fig. 1. The shapes of the deformed membranes are described by $\mathbf{r}_{i}(\theta,\phi)=R_i\mathbf{\hat{r}}+\mathbf{u}(\theta,\phi)$. Where $\mathbf{\hat{r}}$ is the unit vector in the r direction and u is the displacement vector, of which the components are: $u_r = 1/2 s_i (3\cos^2 \theta -1 )$ and $u_{\theta} = - s_i \cos \theta \sin \theta$ [24], satisfying the requirement of local area conservation. This vector describes the deformation as oblate or prolate. Throughout this work the spontaneous curvature is assumed to be zero. If s_i > 0 the deformation is prolate, if s_i < 0 the deformation is oblate, consistent with our agreement on the direction of the field and membrane deformation explained above. The conductivity and dielectric properties of our CLS model are indicated in fig. 1. We assume small membrane deformations and compute the electric fields assuming spherical shapes at all times. In this work, magnetic fields are neglected and the wavelengths used are much smaller than the CLS characteristic size, therefore, we use the quasi-static approximation of the Maxwell equations, $\nabla \times \mathbf{E} =0$.

In general, the electric field has the form

$$\begin{equation} \mathbf{\mathcal{E}}_k(r,t)=\frac{1}{2} \mathbf{E} (r)_ke^{-i\omega t}+\text{c.c.}, \end{equation} \tag{ 2 }$$

where $k=1, 2, \ldots,5$. The Laplace equations for all 5 regions in fig. 1 can be solved to find the electric potentials with the following boundary conditions:

$$\begin{equation} E_{jt}=E_{(j+1)t}, \qquad E_{jn}=\beta_{j} E_{(j+1)n}, \end{equation} \tag{ 3 }$$

where $j=1,2,3,4$, with t and n corresponding to the tangential and the normal components of the electric field and $\beta_{j}=(\sigma_{j+1}-i\omega \varepsilon_{j+1})/(\sigma_{j}-i\omega \varepsilon_{j})$. The electric potential has the well known form: $\varphi_k = ( a_k r + \frac{b_k}{r^2} ) \cos \theta$, so the time-independent electric field can be expressed as:

$$\begin{equation} \mathbf{E}(r,\theta)_k =\tau (r)_k \cos \theta \mathbf{\hat{r}} + \eta (r)_k \sin \theta \mathbf{\hat{\theta}}, \quad k=1,\ldots,5, \quad \end{equation} \tag{ 4 }$$

where the functions $\tau_{k}$ and $\eta_{k}$ are given by

$$\begin{equation} \tau (r)_k =\frac{2 b_k}{r^3} - a_k, \qquad \eta (r)_k = \frac{ b_k}{r^3} + a_k \end{equation} \tag{ 5 }$$

the a's and b's constants are obtained by using the boundary conditions (eq. (3)).

The Maxwell stress tensor is calculated in order to compute the force densities over the membranes. The cases of interest are those for stationary deformations induced by time-independent terms, which can be obtained by averaging over a period of $2 \pi /\omega$ [26]. The force densities are given by

$$\begin{equation} \mathbf{f}_{i} = - \mathbf{n} \cdot (\langle{ T_{i +1}(r_i,\theta)}\rangle - \langle{T_i(r_i,\theta)}\rangle ),\quad i=1,\ldots,4, \end{equation} \tag{ 6 }$$

the bracket denotes average and the dot represents the product between a tensor and a vector. The work generated by the force densities over the membranes can be calculated using

$$\begin{equation} W_i =\int \mathbf{f}_{i} \cdot \mathbf{u }\,\text{d}A_{i} + \int \mathbf{f}_{i+1} \cdot \mathbf{u} \, \text{d}A_{i+1}, \quad i=1,3, \end{equation} \tag{ 7 }$$

where

$$\begin{equation} \mathbf{f}_{i} \cdot \mathbf{u } =\langle{T_{i11} - T_{(i+1)11}}\rangle u_r + \langle{T_{i12}- T_{(i+1)12}}\rangle u_{\theta} \end{equation} \tag{ 8 }$$

and $\text{d}A_{i} = r_{i}^2 \sin \theta \text{d}\theta \text{d}\phi$. Integrating over all the membrane surface we obtain for the nucleus and plasma membrane:

$$\begin{equation} W_{i} = \frac{2\pi}{15}s_{i}E_0^2\left\{ r_{i}^2 \Gamma_{i} + r^2_{i+1} \Gamma_{i+1} \right\}, \end{equation} \tag{ 9 }$$

where E₀ is the amplitude of the external electric field, and $\Gamma_i$ is

$$\begin{eqnarray} \Gamma_i &= \varepsilon_i \left[ | \tau_i (r_i) |^2 + | \eta_i (r_i) |^2 \right] \nonumber\\ & - \varepsilon_{i+1} \left[ | \tau_{i+1} (r_i) |^2 + | \eta_{i+1} (r_i) |^2 \right] \nonumber\\ & + \varepsilon_{i+1} \left[ \tau_{i+1} (r_i) \eta^{*}_{i+1} (r_i) + \tau^{*}_{i+1} (r_i) \eta_{i+1} (r_i) \right] \nonumber\\ & - \varepsilon_i \left[ \tau_i (r_i) \eta^{*}_i (r_i) + \tau^{*}_i (r_i) \eta_i (r_i) \right], \quad i=1,3. \quad \end{eqnarray} \tag{ 10 }$$

The electric properties of the media, membranes, and the frequency of the electric field are in the $\Gamma_{i}$ factors. The total free energy of the membranes that are subjected to an AC electric field are given by

$$\begin{equation} F_i=\Delta F_{be(i)}-W_i. \end{equation} \tag{ 11 }$$

Minimization of the free energy yields the degree of deformation in our CLS, which is

$$\begin{equation} s_i = \frac{R_i E_0^2}{144\left( 1- \frac{M_{sp} R_i}{6} \right) \kappa_i}\left\{ r_{i}^2 \Gamma_i + r^2_{i+1} \Gamma_{i+1} \right\} \end{equation} \tag{ 12 }$$

From eq. (12) a morphological diagram of deformations (MDD) can be generated in the conductivity-frequency space. Transitions are identified by inspection of the changes in sign of s_i. Theoretically, we can manipulate the conductivity of the media, that includes the electrical conductivity of the membranes and the inner medium. By setting $\sigma_{5}=\sigma_{4}=\sigma_{3}$, $\epsilon_{5}=\epsilon_{4}=\epsilon_{3}$ and $r_{5}=r_{4}=r_{3}$, we reduce our CLS to a single bilayer system. By using experimental values reported in the literature for GUVs deformations [26–28], we obtain the same morphological diagram reported in these publications. The MDD for these conditions is represented by the external membrane of the ellipsoid, see fig. 2. When r₅ < r₄ < r₃, we have two concentric membranes in our system. Both membranes are illustrated in fig. 2, where the inner membrane does no show deformations for x > 1 and only oblate deformation for x < 1. Here $x= \sigma_{3} / \sigma_{1}$.

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To obtain a MDD for parameters in physiological conditions we used values reported for cells [29,30] and are given by: $E_0=20\ \text{kV/m}$, $M_{spm}=M_{spn}=0$, $k_m=1.9\times 10^{-19}\ \text{J}$, $\varepsilon_1=\varepsilon_3=\varepsilon_5=7\times10^{-10}\ \text{F/m}$, $\varepsilon_2=\varepsilon_4=1.7\times10^{-7}\ \text{F/m}$, $\sigma_{2}=\sigma_{4}=1.0\times 10^{-7}\ \text{S/m}$, $\sigma_{1}=1.2\ \text{S/m}$, $\sigma_{3}=0.5\ \text{S/m}$, $k_n=1.9\times10^{-19}\ \text{J}$, $r_1=10\ \mu \text{m}$, $r_3=5\ \mu \text{m}$, and $l_n=l_m=4\ \text{nm}$. The membrane deformations will depend on the field frequency and the ratio of the conductivities. We define the external and internal conductivity ratios as $x_{ext}=\sigma_{1}/\sigma_{3}$ and $x_{int}=\sigma_{5}/\sigma_{3}$, respectively. We start our analysis by taking the experimental values $\sigma_3 =0.5\ \text{S/m}$, and $\sigma_5 =1.2\ \text{S/m}$, for which $x_{int}= 2.4$. Thus, x_int > 1, see fig. 3. The MDD shows a similar behavior for the inner membrane but with an oblate-to-prolate transition at nearly $\omega = 30\ \text{kHz}$, see line transition 2 in fig. 3, and an prolate-to-sphere transition at higher frequencies, line transition 3 in the same figure. A new tansition is predicted for the external membrane that depends on both the electric field frequency and x_ext, line transition 1 same figure. For a CLS model we use the conductivity values of the cytoplasm and inner nucleus reported in the literature [30]. It is observe here that is possible to deform the inner membrane without deform the outer membrane, as seen along the transition line number 1 shown in fig. 3 at high frequencies. Note that if we increase the strength of the conductivities, there is a shift in the frequency transitions as shown in figs. 3 and 4.

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There is little information about nuclear internal conductivity; therefore, it is interesting to explore the types of deformation when the nuclear conductivity varies. For this, we fix the values of the external and intermediate conductivity, $\sigma_1=0.5\ \text{S/m}$ and $\sigma_3=1.2\ \text{S/m}$, respectively, see fig. 4. For x_ext < 1, with the transition shapes for the outer membrane are from oblate-to-prolate when the frequency increases and the inner membrane has an oblate-to-prolate transition at larger frequencies, fig. 4. The inner membrane is deformed on the transition line 1 and the external membrane is not perturbed (see inset (a) in fig. 4). On transition line 2, the outer membrane is deformed prolate and the inner membrane has a spherical shape (see inset (b) in fig. 4). In all cases, the deformation of the inner membrane is larger for soft bending rigidity, as expected from eq. (12).

The membrane deformations of the CLS explored here can be understood based on the work by Yamamoto et al. [26]. In that work the shape of vesicle deformations were determined by the sign of s, as we did here. The amplitude of the deformation, s, is obtained from the minimization of the free energy, eq. (11), where the positive or negative sign depends on the work done by the second order Maxwell stress tensors. The normal and tangential electric fields embedded in the tensor contribute to the tensile and shear force densities over the internal and external surfaces of the membranes, given by eq. (6), which are the driving forces deforming the membranes. The positive or negative value of these forces are determined by the $\beta_{j}$'s coefficients. Therefore, the sign of s will depennd on these factors: media conductivity, permittivity, the frequency of the electric field. In Yamamotos's work, the conductivity of the membrane was 10¹⁰ orders of magnitude smaller than the aqueous solution. The dielectric constant was one order of magnitude smaller than the dielectric constant of the aqueous solution, then $\beta_{1,3}\ll1$ (with $\beta_{1}=(\sigma_{2}-i\omega \varepsilon_{2})/(\sigma_{1,3}-i\omega \varepsilon_{1,3})$). Since $\beta_{1,3}$ increase with the frequency ω, the deformations were analyzed in three regions: low-frequency regime ($|\beta_{1}|/\delta_{m}\ll1$ where $\delta=l_{m}/R_{1}$), intermediate frequency regime, $|\beta_{1}|/\delta_{m} \approx 1$, and high-frequency regime, $|\beta_{1}|/\delta_{m} \gg 1$ and $|\beta_{3}|/\delta_{m} \gg1$. The low-frequency regime was determined by $\omega \ll \delta_{m} \sigma_{3}/\epsilon_{2}$. Using numerical values for a single membrane [26], we obtained approximately $\omega \approx 6\ \text{kHz}$, below which the stable shape of the vesicle is prolate independently of the frequency of the electric field, in agreement with the experimental results [27]. In comparison, in this work, we used conductivity values that can be three orders of magnitude higher than the conductivities used in [26], while the permitivitty of the membranes is four orders of magnitude larger than the permitivites used in [26]. Using these values we obtained a frequency of 12 Hz as the limit to the low-frequency regime, where deformations are independent of the electric field frequency. The intermediate-frequency regime, where $\beta_{1}$ and $\beta_{3}$ are $\approx \delta$, which occurs at $\omega=12\ \text{Hz}$, see fig. 3. Below 12 Hz the prolate deformations are independent of the electric field frequency. This occurs because the electric field is shielded from the internal media or have a weak permeation, and the forces over the external membrane are tangential and a prolate deformation is observed. At frequencies higher than 12 Hz, the high value of the membrane conductivity the displacement currents flow across the membrane, hence a normal component of the electric field appears. The electric field permeates the external membrane and the conductivity of the internal medium reshapes the electric field and the Maxwell stresses. Shear and tensile forces, coming from the Maxwell stresses, deform the membranes. The deformation depend on the conductivity ratios between the inner and external media and the electric field frequency. For x_ext < 1, the tangential force density acts on the electric charges accumulated at the membrane surfaces. This force will point towards the poles of the membranes. The normal force density at the poles of the membranes point towards the exterior of the vesicle. Therefore, the two resulting forces deform the vesicle in a prolate shape. For x_ext > 1, there is a change of direction of the forces. The tangential forces point towards the equator and the normal forces at the poles point towards the interior of the vesicles. Both forces deform the vesicle in an oblate shape. The prolate-to-oblate transition comes from the competition between force densities arising from the interaction between accumulated charges at the membranes and the electric fields. At high frequencies, higher than 1 MHz, charge accumulation at the membranes decreases due to the Maxwell-Wagner relaxation frequency. Therefore, the work done by the Maxwell stresses vanishes and we observe spherical shapes as can be seen in figs. 3 and 4. Note that we plot x_ext vs. ω and in Yamamoto's work is plotted $1/x_{ext}$ vs. ω.

Another important factor in the deformations studied here is the geometry of the system. In the case of a single vesicle, the ratio between the thickness of the membrane and the radius of the vesicle $\delta_{m}=l_{m}/R_{1}$ have an important role in redirecting the electric fields. This implies a competition between geometrical parameters and frequency of the electric field [26]. In this work, the inner vesicle imposes a new constraint on the charge and electric field distribution $\delta_{n}=l_{n}/R_{3}$. For low frequencies, the electric fields are redirected tangential over the inner membrane, producing tangential forces pointing to the equator and a normal electric field produces inward forces, deforming the inner membrane in an oblate shape, see figs. 3 and 4. The normal forces over the external and inner membranes counterbalance each other and only tangential forces pointing toward the poles are applied over the external membrane and an oblate shape of the external membrane is observed. By increasing the frequency, the field permeates more and the normal external forces dominate to deform the external membrane in a prolate shape, see fig. 3. The transition line is established by $\omega \ll \delta_{n} \sigma_{5}/\epsilon_{4}$, which using the numerical values cited above gives ω ≈ 20 kHz., see fig. 4.

If we plot x_int vs. ω, we obtained another morphological diagram, fig. 5. In this case, we varied the internal conductivity of the inner vesicle, we obtained the same transition as the previews morphological diagram for the external vesicle, with a transition line for the inner membrane. This transition depends on the frequency and geometrical factors as explained below.

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The intensity of the electric field is modified from one medium to another by a factor that depends on the boundary conditions expressed in eq. (3), where the strength of the conductivity media plays an important role. The parameters used in this numerical calculation (taking into account the values reported experimentally), give us $\sigma_4 \ll \sigma_3$ and $\varepsilon_3 = \varepsilon_5 \ll \varepsilon_4$, then the ratio of the inductances can be expressed as $\beta_3 = -i \frac{\omega \varepsilon_4}{\sigma_3}$ and $\beta_4 = i \frac{\sigma_5}{\omega \varepsilon_4}$. Taking the limit $r_1, r_2 \rightarrow \infty$ we obtain $\beta_1=\beta_2 =0$. In this case, the equation of the inner membrane deformation takes a simple form:

$$\begin{eqnarray} \begin{array}{lll} \displaystyle\dfrac{s_n}{r_n} &= \dfrac{r_n r_3^2 E_0^2 \varepsilon_4}{64 \kappa_n |z_0|^2}\bigg\{ \Lambda_1 x_{int}^2 + \Lambda_2 x_{int} \\ &+ \Lambda_3 \bar{\omega}^2 + \Lambda_4 \dfrac{(x_{int})^2 }{\bar{\omega}^2} + \Lambda_5 \bigg\}, \end{array} \end{eqnarray} \tag{ 13 }$$

where $\bar{\omega} = \omega \varepsilon_4 / \sigma_3$ and the Λ's are constants that depend only on the geometry of the system:

$$\begin{eqnarray} \begin{array}{lll} \Lambda_1 &= 9 \bar{\varepsilon}(1 + 2 \theta_0^3), \\ \Lambda_2 &= \bar{\varepsilon}\left[ 18(1 + 2 \theta_0^3)(\theta_0^3 + 2) + 36(1- \theta_0^3)(\theta_0^3 -1) \right]\\ \Lambda_3 & = 36 \bar{\varepsilon}(1-\theta_0^3)^2,\\ \Lambda_4 & = 81 \theta_0^2 - 9 (1 + 2 \theta_0^3)^2 - 9 (\theta_0^3 -1)^2 \\ &\quad + 18 (\theta_0^3 -1)(1 + 2 \theta_0^3), \\ \Lambda_5 & = 81 \theta_0^3 - 9(2 + \theta_0^3)^2 -36 (1-\theta_0^3)^2 \\ &\quad - 36(1-\theta_0^3)(2 + \theta_0^3) - 243 \bar{\varepsilon} \theta_0^3, \end{array} \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} & z_0 = 3 \left[ -(\theta_0^3 +2) + \beta_4 (\theta_0^3 -1) \right],\qquad \end{eqnarray} \tag{ 15 }$$

where $\theta_0 = r_4/r_3$ and $\bar{\varepsilon} = \varepsilon_3/ \varepsilon_4 = \varepsilon_5 / \varepsilon_4$. The numerical values of Λ's constant for $r_3 = 5\ \mu \text{m}$ and $r_4 = 4.996\ \mu \text{m}$ are

$$\begin{equation} \begin{array}{lll} \Lambda_1 &=& 0.327714, \\ \Lambda_2 &=& 0.655953,\\ \Lambda_3 & =& 8.40292 \times 10^{-7},\\ \Lambda_4 & =& -0.000103,\\ \Lambda_5 & = &-1.37304, \end{array} \end{equation} \tag{ 16 }$$

and we can find the bifurcation curve given by

$$\begin{equation} \Lambda_1 x_{int}^2 + \Lambda_2 x_{int} + \Lambda_3 \bar{\omega}^2 + \Lambda_4 \frac{(x_{int})^2}{\bar{\omega}^2} + \Lambda_5 = 0. \end{equation} \tag{ 17 }$$

Equation (17) is plotted in fig. 5. This bifurcation curve predicts the behavior of the inner membrane deformation as presented in figs. 3 and 4 (see lines 2). For example, using the numerical value for x_int in fig. 3, shows that the frequency transition is constant. While varying x_int will yield the different frequencies at which the inner membrane changes shape. This is very important experimentally because it allows us to measure the transitions of the inner membrane without knowing the inner vesicle conductivity. Then, using fig. 5 and eq. (17) we can estimate the internal conductivity of the inner vesicle. It is important to note that the transition occurs only if we take into account the thickness of the inner membrane, because the Λ's depend on the ratio of the inner radii.

To test our theory, experiments could be conducted with double bilayer systems of different sizes by, for example, jet-injection or double emulsion techniques. In cells, experiments could be conducted on detached cells. In this system, our theory could provide a way to measure the conductivity of the nucleoplasm by measuring the shape transition of the nucleus. In this work, we neglected the membrane conductivity that can affect the charge distribution and the nature of the force densities over the surfaces, in order to have a better model, we will have to take into account this factor. However, this is a first step towards more complex models to study the deformation of multi-bilayer membranes or of organelles and cells.

We acknowledged financial support from CONACYT through project Nos. 169504 and 440 Fronteras de la Ciencia. We thank A. Ramírez-Saito and M. L. González-González for fruitful discussions. HA-E and KA acknowledge support from NSF (1507730).