Issue 4
Journal for Veterinary Medicine, Biotechnology and Biosafety
Volume
4, Issue 4, December 2018, Pages 19–28
ISSN 2411-3174 (print version) ISSN 2411-0388
(online version)
BIOPHYSICAL MODEL OF
BIOLOGICAL CELL CONDUCTIVITY BASED ON THE MEMBRANE ELECTROPORATION PROBABILITY
Shigimaga V. O. 1, Paliy And. P. 1, Pankova O. V. 1, Paliy Anat. P. 2
1 Kharkiv Petro Vasylenko
National Technical University of Agriculture, Kharkiv, Ukraine, е-mail: biovidoc@gmail.com
2 National Scientific Center ‘Institute of
Experimental and Clinical Veterinary Medicine’, Kharkiv, Ukraine,
е-mail: paliy.dok@gmail.com
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PDF (print version)
Citation for print version: Shigimaga, V. O.,
Paliy, And. P., Pankova, O. V.
and Paliy, Anat. P. (2018) ‘Biophysical
model of biological cell conductivity based on the membrane electroporation
probability’, Journal
for Veterinary Medicine, Biotechnology and Biosafety, 4(4),
pp. 19–28.
Download
PDF (online version)
Citation for online version: Shigimaga, V. O.,
Paliy, And. P., Pankova, O. V.
and Paliy, Anat. P. (2018)
‘Biophysical model of biological cell
conductivity based on the membrane electroporation probability’, Journal for Veterinary Medicine, Biotechnology and
Biosafety. [Online] 4(4), pp. 19–28. Available at:
http://jvmbbs.kharkov.ua/archive/2018/volume4/issue4/oJVMBBS_2018044_019-028.pdf
Summary. The membrane electroporation of a biological cell was well known as a convenient, multipurpose and universal
way of temporarily increasing its permeability in a pulsed electric field (PEF) with certain parameters. The process and result of the
membrane interaction with the PEF is
greatly influenced by its heterogeneous biological structure, which has
both native pores of various sizes and various protein inclusions. This leads
to heterogeneity of the electrophysical properties.
All this ultimately affects the cellular conduction in the PEF,
which is both an indicator and an integral characteristic of the
electroporation process of the membrane as a whole. This process can be modelled, considering the physical properties of the
membrane and the cells, as conductors of the pulsed current. However, to
consider in modelling all the features of the native structure of the membrane
pores, as well as newly formed electropores as a result of interaction with the external PEF is impossible. However, if we apply a probabilistic
approach to the formation of electropores, it becomes
possible to construct an adequate model of electroporation. In this article is
presented the developed biophysical (BP) model of cell conductivity,
constructed on the basis of the electropores
formation probability in a membrane under the influence of a pulse electric
field (PEF). The model assumes that in membrane are
formed electropores of different calibers, the
distribution of which submits to normal Gauss’s law. The integral for the
total conductivity of the electroporated membrane is obtained using the integral equation for the total
current through the electropore membrane and the
equation for its conductivity, including the formation of the electropore probability function. The general view of the electropore
formation probability function is received by solution of
Fokker-Planck’s differential equation. Substitution of this equation
solution to conductivity integral gave the general view
of the conductivity function connecting it with electropore
caliber. A comparison of the constructed probability electroporation BP model
with experimental data on mice oocyte conductivity showed that the main reason
for exponential increase of cell conductivity in increasing electrical field
strength is similar nature of conductivity increase with increasing electropore caliber up to membrane breakdown. The
constructed probability BP model of cell conductivity at membrane
electroporation in increasing PEF agrees with the
experimental data.
Keywords: pulse electric field, increasing strength, electropore caliber, cell membrane, Gauss’s law,
conductivity integral
References:
Böckmann, R. A.,
De Groot, B. L., Kakorin, S.,
Neumann, E. and Grubmüller, H. (2008)
‘Kinetics, statistics, and energetics of lipid membrane electroporation
studied by molecular dynamics simulations’, Biophysical Journal,
95(4), pp. 1837–1850. http://dx.doi.org/10.1529/biophysj.108.129437
Chang, D. C., Chassy, B. M.,
Saunders, J. A. and Sowers, A. E. (eds.) (1991) Guide to
electroporation and electrofusion. Academic
Press. http://dx.doi.org/10.1016/C2009-0-21564-9
Dehez, F., Delemotte, L.,
Kramar, P., Miklavčič, D.
and Tarek, M. (2014) ‘Evidence of
conducting hydrophobic nanopores across membranes in
response to an electric field’, The Journal of Physical Chemistry C,
118(13), pp. 6752–6757. http://dx.doi.org/10.1021/jp4114865
den Otter, W. K. (2009)
‘Free energies of stable and metastable pores in lipid membranes under
tension’, The Journal of Chemical Physics, 131(20), p. 205101.
http://dx.doi.org/10.1063/1.3266839
Dermol-Cerne, J. and Miklavcic, D.
(2018) ‘From cell to tissue properties—modeling skin
electroporation with pore and local transport region formation’, IEEE
Transactions on Biomedical Engineering, 65(2), pp. 458–468. http://dx.doi.org/10.1109/TBME.2017.2773126
Fernández, M. L., Risk, M. R. and
Vernier, P. T. (2018) ‘Electropore
formation in mechanically constrained phospholipid bilayers’, The
Journal of Membrane Biology, 251(2), pp. 237–245. http://dx.doi.org/10.1007/s00232-017-0002-y
Fuertes, G., Giménez, D.,
Esteban-Martín, S., Sánchez-Muñoz, O. L.
and Salgado, J. (2011) ‘A lipocentric view
of peptide-induced pores’, European Biophysics Journal, 40(4),
pp. 399–415. http://dx.doi.org/10.1007/s00249-011-0693-4
Golberg, A. and Rubinsky, B.
(2010) ‘A statistical model for multidimensional irreversible
electroporation cell death in tissue’, BioMedical
Engineering OnLine, 9(1), p. 13. http://dx.doi.org/10.1186/1475-925X-9-13
Gowrishankar, T. R., Smith, K. C. and
Weaver, J. C. (2013) ‘Transport-based biophysical system models
of cells for quantitatively describing responses to electric fields’, Proceedings
of the IEEE, 101(2), pp. 505–517. http://dx.doi.org/10.1109/JPROC.2012.2200289
Gurtovenko, A. A. and Lyulina, A. S.
(2014) ‘Electroporation of asymmetric phospholipid membranes’, The
Journal of Physical Chemistry B, 118(33), pp. 9909–9918. http://dx.doi.org/10.1021/jp5028355
Hoiles, W., Krishnamurthy, V. and
Cornell, B. (2018) Dynamics of Engineered Artificial Membranes and
Biosensors. Cambridge University Press. http://dx.doi.org/10.1017/9781108526227
Hoiles, W., Krishnamurthy, V., Cranfield, C. G. and Cornell, B. (2014)
‘An engineered membrane to measure
electroporation: effect of tethers and bioelectronic
interface’, Biophysical Journal, 107(6), pp. 1339–1351.
http://dx.doi.org/10.1016/j.bpj.2014.07.056
Kamenshchikov, S. A. (2014) ‘Transport
catastrophe analysis as an alternative to a monofractal
description: theory and application to financial crisis time series’, Journal
of Chaos, 2014, p. 346743. http://dx.doi.org/10.1155/2014/346743
Kolesnikova, A. A., Shigimaga, V. A.
and Smolyaninova, E. I. (2013) ‘The
influence of stimulation of maturation on electroconductivity
and fertilization rate of mouse oocytes’ [Vliyanie
stimulyatsii sozrevaniya na elektroprovodnost’ i oplodotvoryaemost’ ootsitov myshi], Fundamental
Research [Fundamental’nye issledovaniya],
4(4), pp. 896–899. Available at: https://elibrary.ru/item.asp?id=18860632. [in
Russian]
Kotnik, T., Kramar, P.,
Pucihar, G., Miklavčič, D.
and Tarek, M. (2012) ‘Cell membrane
electroporation—Part 1: the phenomenon’, IEEE Electrical
Insulation Magazine, 28(5), pp. 14–23. http://dx.doi.org/10.1109/MEI.2012.6268438
Kotnik, T., Pucihar, G.
and Miklavčič, D. (2010)
‘Induced transmembrane voltage and its correlation with
electroporation-mediated molecular transport’, The Journal of Membrane
Biology, 236(1), pp. 3–13. http://dx.doi.org/10.1007/s00232-010-9279-9
Kramar, P., Delemotte, L.,
Maček Lebar, A., Kotulska, M., Tarek, M.
and Miklavčič, D. (2012)
‘Molecular-level characterization of lipid membrane electroporation using
linearly rising current’, The Journal of Membrane Biology,
245(10), pp. 651–659. http://dx.doi.org/10.1007/s00232-012-9487-6
Kramar, P., Miklavčič, D.
and Maček Lebar, A.
(2007) ‘Determination of the lipid bilayer breakdown voltage by means of
linear rising signal’, Bioelectrochemistry,
70(1), pp. 23–27. http://dx.doi.org/10.1016/j.bioelechem.2006.03.022
Krassowska, W. and Filev, P. D.
(2007) ‘Modeling electroporation in a single cell’, Biophysical
Journal, 92(2), pp. 404–417. http://dx.doi.org/10.1529/biophysj.106.094235
Luitel, P., Schroeter, D. F.
and Powell, J. W. (2007) ‘Self-electroporation as a model for
fusion pore formation’, Journal of Biomolecular
Structure and Dynamics, 24(5), pp. 495–503. http://dx.doi.org/10.1080/07391102.2007.10507138
Mahnič-Kalamiza, S., Miklavčič, D.
and Vorobiev, E. (2014) ‘Dual-porosity
model of solute diffusion in biological tissue modified by
electroporation’, Biochimica et Biophysica Acta
(BBA) - Biomembranes, 1838(7),
pp. 1950–1966. http://dx.doi.org/10.1016/j.bbamem.2014.03.004
Miklavčič, D. (2012) ‘Network for
development of electroporation-based technologies and treatments: COST TD1104’, The Journal of Membrane Biology,
245(10), pp. 591–598. http://dx.doi.org/10.1007/s00232-012-9493-8
Miklavčič, D. (ed.) (2017) Handbook of
electroporation. Cham, Switzerland: Springer International Publishing. http://dx.doi.org/10.1007/978-3-319-32886-7
Miklavčič, D. and Towhidi, L.
(2010) ‘Numerical study of the electroporation pulse shape effect on
molecular uptake of biological cells’, Radiology and Oncology,
44(1), pp. 34–41. http://dx.doi.org/10.2478/v10019-010-0002-3
Morshed, B. I., Shams, M. and Mussivand, T. (2013) ‘Deriving an electric
circuit equivalent model of cell membrane pores in electroporation’, Biophysical
Reviews and Letters, 08(01n02), pp. 21–32. http://dx.doi.org/10.1142/S1793048012500099
Neu, W. K. and Neu, J. C.
(2009) ‘Theory of Electroporation’, in Efimov, I. R.,
Kroll, M. W. and Tchou, P. J.
(eds.) Cardiac Bioelectric Therapy. Boston, MA: Springer US,
pp. 133–161. http://dx.doi.org/10.1007/978-0-387-79403-7_7
Pavlin, M., Kotnik, T.,
Miklavčič, D., Kramar, P.
and Maček Lebar,
A. (2008) ‘Chapter Seven Electroporation of Planar Lipid Bilayers and
Membranes’, in Leitmannova Liu, A. (ed.) Advances
in Planar Lipid Bilayers and Liposomes, 6. Elsevier,
pp. 165–226. http://dx.doi.org/10.1016/S1554-4516(07)06007-3
Polak, A., Tarek, M.,
Tomšič, M., Valant, J.,
Ulrih, N. P., Jamnik, A.,
Kramar, P. and Miklavčič, D.
(2014) ‘Electroporation of archaeal lipid
membranes using MD simulations’, Bioelectrochemistry,
100, pp. 18–26. http://dx.doi.org/10.1016/j.bioelechem.2013.12.006
Pucihar, G., Krmelj, J.,
Reberšek, M., Napotnik, T.
and Miklavčič, D. (2011)
‘Equivalent pulse parameters for electroporation’, IEEE
Transactions on Biomedical Engineering, 58(11), pp. 3279–3288. http://dx.doi.org/10.1109/TBME.2011.2167232
Rems, L. and Miklavčič, D.
(2016) ‘Tutorial: electroporation of cells in complex materials and
tissue’, Journal of Applied Physics, 119(20), p. 201101. http://dx.doi.org/10.1063/1.4949264
Rems, L., Tarek, M.,
Casciola, M. and Miklavčič, D.
(2016) ‘Properties of lipid electropores II:
comparison of continuum-level modeling of pore conductance to molecular
dynamics simulations’, Bioelectrochemistry,
112, pp. 112–124. http://dx.doi.org/10.1016/j.bioelechem.2016.03.005
Risken, H. (1996) The Fokker-Planck
equation: methods of solution and applications. 2nd ed.
Berlin, Heidelberg: Springer (Springer Series in Synergetics,
18). http://dx.doi.org/10.1007/978-3-642-61544-3
Saulis, G., Saule, R.,
Bitinaite, A., Zurauskiene, N.,
Stankevic, V. and Balevicius, S.
(2013) ‘Theoretical analysis and experimental determination of the
relationships between the parameters of the electric field pulse required to electroporate the cells’, IEEE Transactions on Plasma
Science, 41(10), pp. 2913–2919. http://dx.doi.org/10.1109/TPS.2013.2276918
Schmeer, M., Seipp, T.,
Pliquett, U., Kakorin, S.
and Neumann, E. (2004) ‘Mechanism for the conductivity changes caused
by membrane electroporation of CHO cell-pellets’, Physical Chemistry
Chemical Physics, 6(24), pp. 5564–5574. http://dx.doi.org/10.1039/b411037d
Shigimaga, V. A. (2013a)
‘Nonlinear electric model of biological cell conductivity’ [Nelineynaya elektricheskaya
model’ provodimosti biologicheskoy
kletki], Technical Electrodynamics [Tekhnichna elektrodynamika], 6, pp. 30–35. Available at: http://nbuv.gov.ua/UJRN/TED_2013_6_9. [in Russian]
Shigimaga, V. A. (2013b)
‘Pulsed conductometer for biological cells and
liquid media’, Measurement Techniques, 55(11),
pp. 1294–1300. http://dx.doi.org/10.1007/s11018-013-0124-2
Shigimaga, V. A. (2014) Biotechnical
complex of pulsed conductometry and electromanipulation with animal cells [Biotekhnichnyi
kompleks impulsnoi konduktometrii i elektromanipuliatsii z klitynamy tvaryn]. The dissertation thesis for the scientific
degree of the doctor of technical sciences. Kharkiv: Petro Vasylenko
Kharkiv National Technical University of Agriculture. [in
Ukrainian]
Shigimaga, V. A. (2015) ‘Pulsed conductometry in a variable electric field: outlook for the
development of measurements’, Measurement Techniques, 57(10),
pp. 1213–1218. http://dx.doi.org/10.1007/s11018-015-0605-6
Shigimaga, V. A. (2017) ‘Differential
pulsed conductometer for measurements of the conductivity
of biological cells’, Measurement Techniques, 60(7),
pp. 746–750. http://dx.doi.org/10.1007/s11018-017-1265-5
Shigimaga, V. A., Levkin, D. A.
and Megel, Yu. Ye. (2011)
‘Mathematical modeling of membrane in connection with cell conductivity
in various solutions’ [Matematicheskoe modelirovanie membrany v svyazi s provodimost’yu kletki v razlichnykh rastvorakh], Eastern-European Journal of Enterprise
Technologies [Vostochno-Evropeyskiy zhurnal peredovykh tekhnologiy], 4(4), pp. 53–55. Available at:
http://journals.uran.ua/eejet/article/view/4906.
[in Russian]
Shigimaga, V. A. and Megel, Yu. Ye.
(2011a) ‘Method for determination of the
conductivity of oocytes and embryos in different conditions of the dielectric
medium’ [Metod opredeleniya
provodimosti ootsitov i embrionov v razlichnykh
usloviyakh dielektricheskoy
sredy], Bulletin of the National Technical
University ‘Kharkov Polytechnic Institute’ [Vestnik
Natsional’nogo tekhnicheskogo
universiteta ‘Khar’kovskiy
politekhnicheskiy institut’],
9, pp. 140–144. Available at: http://repository.kpi.kharkov.ua/handle/KhPI-Press/10929.
[in Russian]
Shigimaga, V. A. and Megel, Yu. Ye.
(2011b) ‘Research of conductivity of cells at
change of environment osmotic concentration’ [Issledovanie
provodimosti kletok pri izmenenii osmoticheskoy
kontsentratsii sredy], Eastern-European
Journal of Enterprise Technologies [Vostochno-Evropeyskiy
zhurnal peredovykh tekhnologiy], 2(5), pp. 53–55. Available at:
http://journals.uran.ua/eejet/article/view/1792.
[in Russian]
Shigimaga, V. A. and Megel’,
Yu. Ye. (2012) ‘Application of pulse conductometry
method to investigation of electrical characteristics of biological
cells’ [Primenenie metoda
impul’snoy konduktometrii
dlya issledovaniya elektricheskikh kharakteristik biologicheskikh kletok], Proceedings
of the Institute of Electrodynamics of the National Academy of Sciences of
Ukraine [Pratsi Instytutu elektrodynamiky Natsionalnoi akademii nauk Ukrainy],
31, pp. 147–155. Available at: http://dspace.nbuv.gov.ua/handle/123456789/64133 [in Russian]
Shigimaga, V. A., Megel, Yu. Ye., Kovalenko, S. V.
and Kovalenko, S. M. (2017)
‘Modelling and analysis of electroporation parameters of the membrane of
a biological cell in a varied intensity pulsed electric field’ [Modelirovanie i analiz parametrov elektroporatsii membrany biologicheskoy kletki v impul’snom elektricheskom
pole s izmenyaemoy napryazhennost’yu],
Radio Electronics, Computer Science, Control [Radioelektronika,
informatika, upravlenie],
4, pp. 57–65. http://dx.doi.org/10.15588/1607-3274-2017-4-7. [in Russian]
Smith, K. C. and
Weaver, J. C. (2012) ‘Electrodiffusion
of molecules in aqueous media: a robust, discretized description for
electroporation and other transport phenomena’, IEEE Transactions on
Biomedical Engineering, 59(6), pp. 1514–1522. http://dx.doi.org/10.1109/TBME.2011.2180378
Smolyaninova, E. I., Shigimaga, V. A.
and Kolesnikova, A. A. (2009) ‘The
effect of hormone stimulation on morphological and electric parameters of
murine oocytes’ [Vplyv hormonalnoi
stymuliatsii na
morfolohichni ta elektrychni
parametry ootsytiv myshi], The Animal Biology [Biolohiia
tvaryn], 11(1–2), pp. 328–337.
Available at: http://archive.inenbiol.com.ua:8080/bt/2009/8/4.pdf. [in Ukrainian]
Smolyaninova, Е. I., Shigimaga, V.A.,
Strikha, O. A., Popivnenko, L. I.
and Gordienko, Е. А. (2014)
‘Electric conductivity as a diagnostic parameter for mammalian oocyte and
embryo quality estimation in biotechnology operations’ [Elektricheskaya provodimost’
kak diagnosticheskiy parametr otsenki kachestva ootsitov i embrionov mlekopitayushchikh
v biotekhnologicheskikh operatsiyakh],
Biophysics of Living Cells [Biofizika zhivoy kletki], 10,
pp. 193–195. Available at: https://updoc.site/download/28-30-2014--5addc1754df83_pdf.
[in Russian]
Son, R. S., Smith, K. C.,
Gowrishankar, T. R.,
Vernier, P. T. and Weaver, J. C. (2014) ‘Basic
features of a cell electroporation model: illustrative behavior for two very
different pulses’, The Journal of Membrane Biology, 247(12),
pp. 1209–1228. http://dx.doi.org/10.1007/s00232-014-9699-z
Suzuki, D. O. H.,
Ramos, A., Ribeiro, M. C. M., Cazarolli, L. H.,
Silva, F. R. M. B., Leite, L. D.
and Marques, J. L. B. (2011) ‘Theoretical and experimental
analysis of electroporated membrane conductance in
cell suspension’, IEEE Transactions on Biomedical Engineering,
58(12), pp. 3310–3318. http://dx.doi.org/10.1109/TBME.2010.2103074
Talele, S. and Gaynor, P. (2010)
‘Modelling control of pore number and radii distribution in single-cell
electroporation’, in Elleithy, K., Sobh, T., Iskander, M., Kapila, V., Karim, M. A. and Mahmood, A. (eds.) Technological
developments in networking, education and automation. Dordrecht: Springer,
pp. 231–236. http://dx.doi.org/10.1007/978-90-481-9151-2_40
Towhidi, L., Kotnik, T.,
Pucihar, G., Firoozabadi, S. M. P.,
Mozdarani, H. and Miklavčič, D.
(2008) ‘Variability of the minimal transmembrane voltage resulting in
detectable membrane electroporation’, Electromagnetic Biology and
Medicine, 27(4), pp. 372–385. http://dx.doi.org/10.1080/15368370802394644
Wang, M., Orwar, O.,
Olofsson, J. and Weber, S. G. (2010)
‘Single-cell electroporation’, Analytical and Bioanalytical Chemistry, 397(8), pp. 3235–3248.
http://dx.doi.org/10.1007/s00216-010-3744-2
Yarmush, M. L., Golberg, A.,
Serša, G., Kotnik, T.
and Miklavčič, D. (2014)
‘Electroporation-based technologies for medicine: principles,
applications, and challenges’, Annual Review of Biomedical Engineering,
16(1), pp. 295–320. http://dx.doi.org/10.1146/annurev-bioeng-071813-104622
Ziegler, M. J. and
Vernier, P. T. (2008) ‘Interface water dynamics and porating electric fields for phospholipid bilayers’, The
Journal of Physical Chemistry B, 112(43), pp. 13588–13596. http://dx.doi.org/10.1021/jp8027726
