# Effect of electric field induced transmembrane potential on spheroidal cells: theory and experiment

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## 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.

## Keywords

Chinese hamster ovary cells Electroporation Finite-element modelling Spheroidal cells Transmembrane potential## Abbreviations

## Geometry

*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

## Electric

*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

## Notes

### 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 QLK3-1999-00484) under the framework of the 5th PRCD of the European Commission.

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