Revealing electrical stresses acting on the surface of protoplast cells under electric field
Introduction
Significant progresses have been made in mammalian cell engineering over past years for the modification of cellular function such as gene expression [1], [2], [3], [4], [5], [6], [7], [8], protein processing [9], [10], [11], [12], secretion [13], [14], [15], glycosylation [16], [17] and proliferation [18], [19], [20], [21]. One of the most important aims of cell engineering is to improve the cellular properties of cells for applications in cell therapies and tissue engineering. Electroporation is one of the promising approaches to transfer macromolecule into cell. It allows one to introduce exogenous molecules into cells and simultaneously to extract endogenous molecules from inside of the cells [22]. The formation of hydrophilic pores on the surface of cell membrane makes the cell permeable for achieving this process. The pores are formed by the tension stress exerted on the cell membrane, either mechanically (pipet aspiration) or electrically (electroporation). Up to now, the formation mechanism of the pores caused by electrocompressive stresses has not yet been understood well. It is generally believed that the transmembrane potential reaches a critical value to make cell to rupture. The rupture or irreversible membrane breakdown happens when the pore radius becomes more than a critical value [23]. Membrane rupture mechanism under electric field is also unclear like membrane permeabilization. Electrical stresses should have an important role in both phenomena.
Different theoretical models have been proposed to explain this electroporation phenomenon. The transient aqueous model is one of the most accepted models. It assumed that when cell is exposed to an electric field, the induced transmembrane potential (TMP) provides free energy for phospholipids orientation in the cell membrane, leading to the formation of hydrophilic pores [24], [25], [26]. In a previous study, the effect of electric field on cell membrane permeability has been investigated indirectly by measuring the conductivity changes caused by applying electrical pulses and observing molecular transport into cells [27]. TMP depends on cell radius, the electric field intensity, the angle between membrane points and direction of the electric field. It can be calculated by the Schwan’s equation [28]. It has been observed that molecule transfection happens more in the part of membrane where TMP is higher. This could be identified by checking the electrical stress distribution acting on the cell surface. Priya and Gowrisree studied the effects of the electric field on the cell in a conductive environment [29]. In their work, TMP was calculated numerically and compared with analytical values, while electroporation was investigated by measuring the TMP. However, in their research, the effect of electric field on the cell was studied only by calculating the potential difference between the inner and outer surface of membrane. In addition, Tekle et al. studied the formation and distribution of pores as well as its lifetime in the cell membrane by applying the electric field [30]. Lysis, a permanent physical rupture of cells plasma membrane, is another important phenomenon when exposure to electric field. The membrane will rupture if the area density of electrical energy reaches certain critical value(typically 5 mJ/m2) and the transmembrane potential is about 0.51.0 V, depending on the duration of applied field [31]. On the other hand, lysis may also be induced by mechanical stresses. Cell membrane will rupture if the surface tension reaches required value (typically 5 mN/m) [31]. This finding indicates that the mechanism of cell membrane rupture can be studied by monitoring the electrical stresses and strains produced over cell surface in the presence of electric field.
In addition, electro-mechanical permeabilization of lipid vesicles have been investigated by Needham and Hochmuth [32]. They used a micropipette to apply mechanical tension on the cell and studied the effect of mechanical stresses and electrocompressive stresses on cell permeabilization. The critical electric field intensity was provided as a function of mechanical tension on the membrane. However, they did not discuss the effect of the electric field on the cell. Although membrane rupture and pores formation in phospholipid bilayers under the effect of electrical and mechanical stresses have been investigated by molecular dynamics, the mechanism of pores formation in the cell membrane under an electric field is still unclear [33]. In another study, Akinlaja & Sachs reported the combined effects of mechanical and electrical stress on membrane breakdown [34]. It claimed that the mechanism for this membrane breakdown remain unclear. In their experimental research, the effect of electric field on cell membrane was studied indirectly by measuring the conductance and capacitance. The mechanical tension exerted on the membrane was approximated by cell-attached patches from the applied pressure.
As mentioned above, the mechanism of pore formation in the cell membrane when the cell is exposed to an electric field has not been clearly identified. It has mostly been studied indirectly by measurements of conductivity of cell suspensions and cell pellets [35], electro-optical experiments [36] and some other techniques. But the structural changes in the cell membrane in the presence of an electric field have not been studied directly. To better understand the electroporation phenomenon and cell membrane permeability, it was necessary to closely monitor the interaction between cells and electric field. Researchers have examined the dielectrophoretic effect on cells to investigate electroporation phenomena [37]. The effects of dielectrophoresis force in DNA accumulation around the cell membrane has been studied [28]. It has been reported that applying high-voltage electric pulse made the cell membrane permeable while applying lower voltage pulse helped transfer DNA into the cell. Nevertheless, because its dielectric properties were different from insulating suspension fluid, when a cell was exposed to an electric field caused local changes in the electrostatic potential distribution and local non-uniformity in electric field. As a result, electrical stresses were applied on cell surface. The electrical stresses may play a key role in cell membrane structural changes and its permeability which can never be underestimated. These electrical stresses are sometimes known as electrocompressive stresses in literature but depending on cell and insulating suspension fluid dielectric properties, the electrical stress distribution over cell surface is either tensile or compressive.
In this paper, we employ the numerical simulation to study the electrical stresses generated on a cell when it is exposed to an electric. We also discuss the affecting factors that have considerable influence on the distribution and intensity of electrical stresses. Immersed interface method (IIM) is selected and developed to solve the governing equations for this problem based on electrostatic question and the internal boundary conditions, and the electric field is calculated along the outer and inner boundaries of cell membrane. The electric stresses exerted on the cell surface are calculated using Maxwell stress tensor. The model is verified by the numerical results obtained via effective dipole moment (EDM) approach which has been widely used and validated by previous experiments. Specifically, the DEP force induced on a cell that is exposed to a nonuniform electric field is calculated by integrating Maxwell stress tensor over cell surface. However, it is noted that EDM approximation becomes less accurate when the size of cell is not negligible in comparison with the size of device. Thus, to validate the model, the cell radius should be smaller compared to the device size so that the differences between the results become negligible.
Section snippets
Theory and governing equations
In a variety of biomedical applications, cells are exposed to electric fields. When an electric field applies on colloidal particles suspended in a fluid, the major electrokinetic phenomena generated include electrophoresis (EP), induced-charge electrophoresis (ICEP) and dielectrophoresis (DEP). When an electric field is applied on cells suspended in a fluid, the most important phenomenon is DEP, which has many applications such as cell trapping, sorting and separating bioparticles [38]. It is
Numerical method and solutions
In the past decades, there are a few numerical methods proposed for solving partial differential equations (PDE) which have irregular boundaries in the solution domain and discontinuous coefficients such as smoothing method [45], harmonic averaging [46], Peskin’s immersed boundary (IB) method [47] and immersed interface method (IIM) [48]. In these methods the coefficients of equations are usually discontinuous, i.e., they have jump interface conditions or singular force conditions on the
Results and discussion
Convergence analysis was carried out first for the numerical solutions from fast IIM. To examine this issue, the problem has been solved with different mesh sizes and the results are presented in Fig. 2, in which the electrostatic potential at the line is described. It should be noted that the number of control points and the number of points used for interpolation are also important for the convergence of fast IIM. In the calculation, the number of points for the interpolation is 16 and
Conclusion
In this study, we have numerically investigated the interaction of a protoplast cell and electric field. The governing PDEs equations are solved by fast IIM method with interfaces and irregular domains involved. Results showed that the presence of cell would distort the local distribution of electric potential and electric field, and these changes were dependent on the relative permittivity between cell and insulating suspension fluid. The distribution of electrical stresses acted over the cell
Acknowledgments
The work was supported by Iran National Science Foundation. The work was also partially supported by the Startup fund from the School of Packaging at Michigan State University, USA .
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