Abstract
The transmembrane potential on a cell exposed to an electric field is a critical parameter for successful cell permeabilization. In this study, the effect of cell shape and orientation on the induced transmembrane potential was analyzed. The transmembrane potential was calculated on prolate and oblate spheroidal cells for various orientations with respect to the electric field direction, both numerically and analytically. Changing the orientation of the cells decreases the induced transmembrane potential from its maximum value when the longest axis of the cell is parallel to the electric field, to its minimum value when the longest axis of the cell is perpendicular to the electric field. The dependency on orientation is more pronounced for elongated cells while it is negligible for spherical cells. The part of the cell membrane where a threshold transmembrane potential is exceeded represents the area of electropermeabilization, i.e. the membrane area through which the transport of molecules is established. Therefore the surface exposed to the transmembrane potential above the threshold value was calculated. The biological relevance of these theoretical results was confirmed with experimental results of the electropermeabilization of plated Chinese hamster ovary cells, which are elongated. Theoretical and experimental results show that permeabilization is not only a function of electric field intensity and cell size but also of cell shape and orientation.
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Abbreviations
 a :

length of box side
 d :

membrane thickness
 g _{ϕϕ} and g _{ττ} :

elements of metric tensor in spheroidal coordinates
 p(ϑ):

arc length on an ellipse
 r _{ i } :

vector of the point T(x,y,z) lying at the surface of the spheroid
 R :

cell radius in the case of a sphere
 R _{1}, R _{2}=R _{3} :

axes of cell in the case of a spheroid (prolate spheroid: R _{1}>R _{2}=R _{3}; oblate spheroid R _{1}<R _{2}=R _{3})
 S :

area (surface)
 T :

point at the surface of the spheroid
 ρ:

ratio between R _{1} and R _{2}
 ϑ, ϕ, r :

spherical coordinates
 τ, ϕ, σ:

spheroidal coordinates
 E :

applied electric field
 j _{n} :

normal component of electric current at the surface of the spheroid
 L _{ i } :

depolarizing factor in the i=x, y and z directions
 α:

azimuth angle in the spherical coordinate system; the angle between the symmetry axis of the spheroid and the external electric field
 β:

polar angle in the spherical coordinate system
 φ:

electric potential
 σ_{i} :

conductivity of cytoplasm
 σ_{o} :

external medium conductivity
 σ_{m} :

conductivity of cell membrane
 Δφ:

transmembrane potential
 Δφ_{c} :

threshold or critical transmembrane potential
 Δφ_{i} :

induced transmembrane potential
 Δφ_{r} :

resting transmembrane potential
References
Bernhardt J, Pauly H (1973) On the generation of potential differences across the membranes of ellipsoidal cells in an alternating electrical field. Biophysik 10:89–98
Bier M, Hammer SM, Canaday DJ, Lee RC (1999) Kinetics of sealing for transient electropores in isolated mammalian skeletal muscle cells. Bioelectromagnetics 20:194–201
Brock TD, Smith DW, Madigan MT (1984) Biology of microorganisms. PrenticeHall, Engelwood Cliffs, NJ, USA
Eynard N, Rodriguez F, Trotard J, Teissié J (1998) Electrooptics studies of Escherichia coli electropulsation: orientation, permeabilization, and gene transfer. Biophys J 75:2587–2596
Fear CE, Stuchly MA (1998) Modeling assemblies of biological cells exposed to electric eields. IEEE Trans Biomed Eng 45:1259–1271
Fricke H (1925) A mathematical treatment of the electrical conductivity and capacity of disperse systems. II. The capacity of conducting spheroids surrounded by a nonconducting membrane for a current of low frequency. Phys Rev 26:678
Gabriel B, Teissié J (1998) Fluorescence imaging in the millisecond time range of membrane electropermeabilisation of single cells using a rapid ultralowlight intensifying detection system. Eur Biophys J 27:291–298
Gimsa J, Wachner D (2001) Analytical description of the transmembrane voltage induced on arbitrarily oriented ellipsoidal and cylindrical cells. Biophys J 81:1888–1896
Hart FX, Marino AA (1982) ELF dosage in ellipsoidal model of men due to high voltage transmission lines. J Bioelec 1:129–154
Korn GA, Korn TM (2000) Mathematical handbook for scientists and engineers. Dover, Mineola, NY, USA
Kotnik T, Miklavčič D (2000) Analytical description of transmembrane voltage induced by electric fields on spheroidal cells. Biophys J 79:670–679
Kotnik T, Bobanović F, Miklavčič D (1997) Sensitivity of transmembrane voltage induced by applied electric fields: a theoretical analysis. Bioelectrochem Bioenerg 43:285–291
Krasznai Z, Marian T, Balkay L, Emri M, Tron L (1994) Flow cytometric determination of absolute membrane potential of cells. J Photochem Photobiol 28:93–99
Miklavčič D, Šemrov D, Mekid H, Mir LM (2000) A validated model of in vivo electric field distribution in tissues for electrochemotherapy and for DNA electrotransfer for gene therapy. Biochim Biophys Acta 1523:73–83
Miller CE, Henriqez CS (1988) Threedimensional finite element solution for biopotentials: erythrocyte in an applied field. IEEE Trans Biomed Eng 35:712–718
Mir LM (2001) Therapeutic perspectives of in vivo cell electropermeabilization. Bioelectrochemistry 53:1–10
Neumann E (1989) The relaxation hysteresis of membrane electroporation. In: Neumann E, Sowers AE, Jordan CA (eds) Electroporation and electrofusion in cell biology. Plenum, New York, pp 61–82
Neumann E, SchaeferRidder M, Wang Y, Hofschneider PH (1982) Gene transfer into mouse lyoma cells by electroporation in high electric fields. EMBO J 1:841–845
Pavlin M, Pavšelj N, Miklavčič D (2002) Dependence of induced transmembrane potential on cell density, arrangement, and cell position inside the cell system. IEEE Trans Biomed Eng 49:605–612
Rols MP, Teissié J (1998) Electropermeabilization of mammalian cells to macromolecules: control by pulse duration. Biophys J 75:1415–1423
Rols MP, Delteil C, Golzio M, Dumond P, Cros S, Teissie J (1998a) In vivo electrically mediated protein and gene transfer in murine melanoma. Nat Biotechnol 16:168–171
Rols MP, Delteil C, Golzio M, Teissié J (1998b) In vitro and ex vivo electrically mediated permeabilization and gene transfer in murine melanoma. Bioelectrochem Bioenerg 47:129–134
Satkauskas S, Bureau MF, Puc M, Mahfoudi A, Scherman D, Miklavcic D, Mir LM (2002) Mechanisms of in vivo DNA electrotransfer: respective contributions of cell electropermeabilization and DNA electrophoresis. Mol Ther 5:133–140
Schwan HP (1957) Electrical properties of tissue and cell suspensions. Adv Biol Med Phys 5:147–209
Schwarz G, Saito M, Schwan HP (1965) On the orientation of nonspherical particles in an alternating electrical field. J Chem Phys 10:3562–3569
Šemrov D, Miklavčič D (1998) Calculation of the electrical parameters in electrochemotherapy of solid tumours in mice. Comput Biol Med 28:439–448
Serša G, Čemažar M, Šemrov D, Miklavčič D (1996) Changing electrode orientation improves the efficacy of electrochemotherapy of solid tumors in mice. Bioelectrochem Bioenerg 39:61–66
Sillars RW (1937) The properties of dielectrics containing semiconducting particles various shapes. J Inst Elec Eng 80:378–394
Stratton JA (1941) Electromagnetic theory. McGrawHill, New York
Sukharev SI, Klenchin VA, Serov SM, Chernomordik LV, Chizmadzhev YA (1992) Electroporation and electrophoretic DNA transfer into cells. The effect of DNA interaction with electropores. Biophys J 63:1320–1327
Susil R, Šemrov D, Miklavčič D (1998) Electric fieldinduced transmembrane potential depends on cell density and organization. Elec Magnetobiol 17:391–399
Teissié J, Blangero C (1984) Direct experimental evidence of the vectorial character of the interaction between electric pulses and cells in cell electrofusion. Biochim Biophys Acta 775:446–448
Teissié J, Rols MP (1993) An experimental evaluation of the critical potential difference inducing cell membrane electropermeabilization. Biophys J 65:409–413
Tekle E, Astumian RD, Chock PB (1990) Electropermeabilization of cell membranes: effect of the resting membrane potential. Biochem Biophys Res Commun 172:282–287
Weaver JC, Chizmadzhev YA (1996) Electroporation. In: Polk C, Postow E (eds) Biological effects of electromagnetic fields. CRC, Boca Raton, Fla., USA, pp 247–274
Weaver JC, Powell KT (1989) Theory of electroporation. In: Neumann E, Sowers AE, Jordan CA (eds) Electroporation and electrofusion in cell biology. Plenum, New York, pp 111–112
Zimmermann U (1982) Electric fieldmediated fusion and related electrical phenomena. Biochim Biophys Acta 694:227–277
Zimmermann U, Pilwat G, Riemann F (1974) Dielectric breakdown of cell membranes. Biophys J 14:881–899
Acknowledgements
The authors thank C. Millot for providing CHO cells. M.G. was supported by a grant from the Association Française contre les Myopathies (AFM). This work was partly supported by the Ministry of Education, Science and Sport of the Republic of Slovenia through various grants and partly by the Cliniporator project (grant QLK3199900484) under the framework of the 5th PRCD of the European Commission.
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Appendix
Appendix
Appendix A
In this section, the analytical solution for the induced transmembrane potential Δφ_{i} on a spheroidal cell is briefly discussed. Under normal physiological conditions the membrane conductance is several orders of magnitude lower than the external medium (Weaver and Chizmadzhev 1996; Kotnik et al. 1997), thus reducing the problem to solving a Laplace equation for a potential φ on a surface of a nonconductive spheroid lying in the external electrical field:
where j _{n} is the normal component of the electric current at the surface of the spheroid, which for a nonconductive membrane is zero. This is analogous to solving a problem of the potential distribution at the surface of a dielectric spheroid (Stratton 1941). This solution can be extended to the frequencydependent problem of a spheroid surrounded by a shell having both dielectric and conductive properties, and has been applied to cells by several authors to calculate the frequencydependent Δφ_{i} (Schwartz et al. 1965; Bernhardt and Pauly 1973; Zimmermann et al. 1974; Gimsa and Wachner 2001). The solution for a DC case can be correspondingly obtained by solving the Laplace equation in the spheroidal coordinate system, which for parallel orientation has been given by Kotnik and Miklavčič (2000).
The generalized Schwan equation for an arbitrary oriented ellipsoid can be written as:
where the r _{ i } is the vector of the point T(x, y, z) lying at the surface of the ellipsoid and L _{ i } are the depolarizing factors in the x, y and z directions and are dependent only on the geometrical properties of the ellipsoid. The sum of the depolarizing factors is always 1.
Analytical equations for depolarizing factors for an ellipsoid can be found in the paper by Stratton (1941). Here we shall limit ourselves only to an axially symmetrical ellipsoid where R _{1}>R _{2}=R _{3} for a prolate spheroid and R _{1}<R _{2}=R _{3} for an oblate spheroid. The depolarizing factor for the prolate spheroid along the symmetry axis is:
and for the oblate spheroid is:
The depolarizing factors in the other two directions can be calculated from the condition that the sum is equal to one:
If the z axis of the coordinate system is parallel to the symmetry axis of the spheroid, then the solution for the parallel and perpendicular orientations is:
For a sphere where R _{1}=R _{2}=R, then L _{ x }=L _{ y }=L _{ z }=1/3 and z=R cosϑ; thus from Eq. 12 we obtain:
which is the wellknown Schwan equation (Eq. 1). See Table 1.
In its most general case the electric field orientation is defined by the angles α (Fig. 1c) and β:
but without loss of generality we can always choose β=0 so the vector of the electric field lies in the xz plane. The point T(x, y, z) at the surface of the spheroid in the spherical coordinates can be written as:
where the absolute value of ∣r∣ is:
Introducing Eqs. 14, 15, 16 into Eq. 8, Δφ_{i} on an arbitrary oriented spheroid for an infinitely small membrane conductance can be obtained:
For β=0 and presented in the Cartesian coordinate system, the above equation simplifies to:
This solution is the same as derived by Gimsa and Wachner (2001), only presented in a form that is analogous to the Schwan equation and already taking into account all simplifications that are valid for the cells. Thus from the solution for the induced potential in parallel and perpendicular orientations, one can calculate Δφ_{i} on an arbitrarily oriented spheroid by means of a linear combination of the two solutions. In other words, in the coordinate system of the spheroid the electric field has two components: in the x and z directions. Since the potential is a linear function, we can simply add the potential induced in the perpendicular direction and the one induced in the parallel direction.
Appendix B
For the calculation of the area where the critical transmembrane potential Δφ_{c} is exceeded, we have to integrate the surface on the spheroid in spheroidal coordinates τ, ϕ and σ (Korn and Korn 2000). The transformation from Cartesian coordinates to spheroidal coordinates for a prolate spheroid is defined by:
and for oblate spheroid by:
The surface element in spheroidal coordinates is (Korn and Korn 2000):
where g _{ϕϕ} and g _{ττ} are elements of the metric tensor in spheroidal coordinates.
The equation of the surface of the spheroid is:
where ϕ_{1}, ϕ_{2}, τ_{1} and τ_{2} are borders of integration defined by the condition that Δφ_{i} = Δφ_{c}. As already mentioned, we analyzed Δφ_{i} on an ellipse, obtained from the crosssection of a spheroid with the xz plane. For two points on the ellipse where Δφ_{c} was exceeded, an ellipse on a spheroid through these two points can also be defined, which is perpendicular to plane xz. This ellipse represents our border of integration. The border of integration where Δφ_{c} has been exceeded is a closed curve on an ellipsoid which is nearly an ellipse. By transforming this ellipse into spheroidal coordinates, the ϕ_{1}, ϕ_{2}, τ_{1} and τ_{2} values are obtained.
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Valič, B., Golzio, M., Pavlin, M. et al. Effect of electric field induced transmembrane potential on spheroidal cells: theory and experiment. Eur Biophys J 32, 519–528 (2003). https://doi.org/10.1007/s0024900302969
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DOI: https://doi.org/10.1007/s0024900302969