This study first considers that voltages of cellular organelle membranes could significanly surpass these of plasma membranes under the influence of ultrashort and high-intensity electric pulse. This is due to the voltages induced on the membrane. Using an approximate theory coordinated with the Kotnik's analytical method, considering the electroporation, we focus on the reactions of cell membranes placed in a trapezoidal pulse. Then, we discuss conductive power dissipations of normal cell and cancer cell generated by a sinusoidal exposure which include dielectric relaxation effects. In comparison with  the complex numerical calculations of Joshi \textit{et al}, our results are in very good agreement

Giang, Nguyen Duc

Nhung, Nguyen Thi Thuy

Thanh, Ngo Van

Tung, Le Thanh

Viet, Nguyen Ai

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 20, No. 3 (2010), pp. 211-218
EFFECTS OF ELECTROPORATION ON BIOLOGICAL
MEMBRANES EXPOSED TO HIGH POTENTIALS
NGUYEN THI THUY NHUNG, LE THANH TUNG, NGUYEN DUC GIANG,
NGO VAN THANH, AND NGUYEN AI VIET
Institute of Physics, VAST
Abstract. This study first considers that voltages of cellular organelle membranes could signif-
icanly surpass these of plasma membranes under the influence of ultrashort and high-intensity
electric pulse. This is due to the voltages induced on the membrane. Using an approximate theory
coordinated with the Kotnik’s analytical method, considering the electroporation, we focus on the
reactions of cell membranes placed in a trapezoidal pulse. Then, we discuss conductive power
dissipations of normal cell and cancer cell generated by a sinusoidal exposure which include di-
electric relaxation effects. In comparison with the complex numerical calculations of Joshi et al,
our results are in very good agreement.
I. INTRODUCTION
Electroporation in biological cell is related with rapid structural rearrangement and
formation of pores in the lipid layer, in respond to an externally applied electric field. It is
leading to an important technique for inserting active molecules into cells and investigating
electrically characteristic differences between normal and cancer cell. This article deals
with the problem with a spherical cell with uniform interior, exposed to electric fields.
The simplest model of a cell with an organelle is then obtained by incorporating into a
cell another spherical body surrounded by a concentric shell [1, 2].
Dealing with this system, Joshi et al (2004) use a total numerical approach [2].
They simulate the spherical computational region including the cell by discretizing it in r
and θ directions. They also calculate the potentials including the dynamic pore model to
deduce the changes of the conductivity of inner and outer membranes during pulse periods.
This phenomenon is called electroporation. The pores are controlled by the Smoluchowski
equation to diffuse across the energy landscape.This equation is so complex that it can
only be solved by numerical method.
In keeping with the same system, Kotnik et al (2006) use the analytical approach in
which they write down the roots of the Poisson quation in spherical coordinates [1]. Then
they use the continuous conditions of the electric potentials Ψ for five regions and for
the normal component of the electric current densities, Λ(∂Ψ/∂r), with Λ is the complex
conductivity of a region. After solving a set of first-order equations, they achieve the
analyical expressions of the transmembrane voltages, ∆Ψm, as the difference between the
212 EFFECTS OF ELECTROPORATION ON BIOLOGICAL MEMBRANES ...
potentials Ψ at the outer and inner surface of this membranes:
∆Ψm = fEacosθ (1)
where E is the strength of the electric field, θ is the polar angle measured with respect
to the direction of the field, and f is a function reflecting the electrical and geometrical
properties of the cell [5]. For the sinusoidal electric fields, Λ becomes σ + jω, where ω
is the angular frequency of the field, σ is the electric conductivity and  is the dielectric
permittivity. For the pulses, they substitute Λ by σ + s, the strength of the electric
field E by the Laplace transformation of the pulse from the time domain. Eventually,
they use an inverse Laplace transformations to achieve the transmembrane voltages in the
time domain, ∆Ψcell(t) = L
−1[∆Ψcell(s)] and ∆Ψorg(t) = L
−1[∆Ψorg(s)]. An other the-
ory approaching electroporation angles is Krassowska’s approximate theory (1999) about
modelling electroporation in a single cell develop an approximated model of electropora-
tion, the model that bases only on an ordinary differential equation (ODE) [3, 4]. The
pore density N (t) is governed by the ODE:
dN
dt
= αe(Vm/Vep)
2
(
1−
N
No
e−q(Vm/Vep)
2
)
. (2)
And the pore current density is N · iep, with the current through single pore [4]:
iep =
pir2mσνmRT
Fh
×
eνm−1(
wo exp(wo − nνm)− nνm
wo − nνm
eνm −
wo exp(wo + nνm) + nνm
wo + nνm
) (3)
where Vm is the transmembrane voltage on the membrane and νm = VmF/RT is the nondi-
mentional transmembrane voltage. Although having a simpler model of electroporation,
they also use the discretizing method in calculation potentials. Values of all paremeters
are given in Table 1
II. METHOD
In this study, we improve the combined method in investigating the properties
transmembrane voltages of cancer cells and normal cell as well as the differences between
them in exposure to the triangular and trapezoidal pulses. We use the analytical equations
of voltages from Kotnik et al and also include the dynamic conductivity properties by the
approximated model of Krassowska et al. All the kinds of cells and pulses are taken from
Joshi’s paper and the results are compared to Joshi to judge the efficiency of the combined
models. We use the continuum model of pore where the pore resistance, Rp = h/(pisr
2),
and the input resistance, Ri = 1/(2sr) are connected in series.
In addition, we expand the analytical calculation of Kotnik et al to solve a problem
of power dissipation of both cells with dielectric relaxation processes. We list here all the
factors of cells and pulses used in this article [2] (Table 2).
NGUYEN THI THUY NHUNG et al. 213
Table 1. Geometric, electrical, and electroporation parameters [3].
Symbol Value Definition
α 100.0 cm−2ms−1 Electroporation parameter
q 2.46 Electroporation constant
Vep 258 mV Characteristic voltage of electroporation
rm 0.76 nm Pore radius
a 50.0 nm Cell radius
h 5.0 nm Membrane thickness
n 0.15 Relative entrance length of the pore
gl 0.19 mS/cm
2 Specific membrane resistance
El 283.75 mV Reversal potential of ionic current
Cm 0.95 mF/cm
2 Specific membrane capacitance
si 4.55 mS/cm Intracellular specific conductivity
se 50.0 mS/cm Extracellular specific conductivity
T 295 K(22oC) Temperature
R 8.314472 Universal gas constant
s 13.0 mS/cm Conductivity of aqueous solution in pores
No 1.5× 105 cm
−2 Equilibrium pore density when Vm = 0 mV
wo 2.65 Energy barrier within pore
III. STEPS AND RESULTS
III.1. Trapezoidal Pulses
Firstly, we use the Laplace-transform method in calculating the transmembrane
voltages of a typical cell exposured to the two kinds of trapezoidal pulses as in the first
column of Table. 2: a trapezoidal pulse with intensity 45 × 105V/m, rise and fall time
10 ns and duration 70 ns.
1.´10-7 2.´10-7 3.´10-7 4.´10-7 5.´10-7 6.´10-7
TimeHsL
1
2
3
4
5
6
7
DYOrg, DYCellHVL
1.´10-7 2.´10-7 3.´10-7 4.´10-7
TimeHsL
0.5
1.0
1.5
DYOrg, DYCellHVL
Fig. 1. The time courses of votages induced
by a trapezoidal pulse of normal cell. Solid:
cell, dashed: organelle
Fig. 2. The time courses of votages induced
by a trapezoidal pulse of cancer cell. Solid:
cell, dashed: organelle
214 EFFECTS OF ELECTROPORATION ON BIOLOGICAL MEMBRANES ...
Table 2. Typical parameters of cells and electric pulses for this analysis [2].
Typical cells Nor B-cells Malig B-cells
Conductivities (S/m)
Evironment 0.6 0.6 0.6
Cell membrane 0.0 5.6× 10−5 9.1× 10−6
Cytoplasm 0.6 1.31 0.48
Nuclear envelope 0.0 1.11× 10−2 4.4× 10−3
Nucleoplasm 0.6 2.04 1.07
Relative permittivity
Evironment 80 80 80
Cell membrane 8 12.8 9.8
Cytoplasm 80 60 60
Nuclear envelope 4 106 60.3
Nucleoplasm 80 120 120
Geomatry parameters (µm)
Cell radius 5 3.3 5.2
Outer mem thickness 0.005 0.007 0.007
Nucleus radius 1 2.8 4.4
Nucleus mem thickness 0.01 0.04 0.04
Figures 1 and 2 depict that the voltages of inner membranes are higher than these
of outer membranes in both cases. Comparing these values with Joshi’s ones, however,
we found that the values of potentials are larger than Joshi’s calculations by a few volts.
This is due to the fact that the analytical calculation of Kotnik does not consider the dy-
namic properties of the conductivity biological of membranes. Due to the electroporation
phenomenon, under an external voltage of above 1.0 V, the conducting pores appear on
membranes, and therefore, increase the effective conductivity of membrane. This leads to
a decrease in the induced voltages on the membranes.
These inadequate results can be made good by using an approximated model of
current through pores (Alan Barnett), mentioned in the paper of Krassowska et al. [4].
We consider here the pore current described by the Eq. (3). The point is simple: we
subtract the voltages induced by the appearances of the pores.
∆Ψce = ∆Ψcell −∆Vm,cell,
∆Ψor = ∆Ψorg −∆Vm,org.
(4)
with ∆Vm,cell and ∆Vm,org is respectively the voltages induced on outer and inner mem-
brane. This voltage can be considered to be the multiplication between the pore current
and the resistance of a single pore. If we consider the pores to be parallel connected,
the overall decrease of voltage can be the decrease of voltage on a single pore. Note that
in the Eq. 3, rm is the radius of pores, in which the pore energy has a local minimum,
rm = 0.8nm, and has a largest probability to appear [3]. The voltages made by pores are
NGUYEN THI THUY NHUNG et al. 215
calculated by the equations:
∆Vm,cell = icell,ep.(Rcell,p +Rcell,i).
∆Vm,org = iorg,ep.(Rorg,p+Rorg,i).
(5)
with icell,ep, iorg,ep are calculated by Eq.3 in which Vm is respectively replaced by ∆Ψcell
and ∆Ψorg Using this update, we re-draw the membranes voltages for two pulses above.
We see here the good agreement between numerical and analytical approaches.
1.´10-7 2.´10-7 3.´10-7 4.´10-7 5.´10-7 6.´10-7
TIME
1
2
3
4
5
6
7
DYOrg, DYCell
Fig. 3. The time courses of votages induced
by a trapezoidal pulse of normal cell.
Dashed: Kotnik’ calculation [1] (red: outter,
purple: inner), thick: our calculation.(green:
outer, dark: inner) Vm.
1.´10-7 2.´10-7 3.´10-7 4.´10-7 5.´10-7 6.´10-7
TIME
0.5
1.0
1.5
DYOrg, DYCell
Fig. 4. The time courses of votages induced
by a trapezoidal pulse of normal cell.
Dashed: Kotnik’ calculation [1] (red: outter,
purple: inner), thick: our calculation.(green:
outer, dark: inner).
Figures 3 and 4 point out that the voltages of inner membranes always surpass these
of outer membranes of both normal and cancer cells when they are exposed to the electric
pulse. Particularly, potentials of cancer membranes (for inner one: about 0.4 V, for outer
one: about 1.2 V) are considerably less than potentials of normal membrane (for inner
one: about 0.7 V, for outer one: about 4.5 V).
The effect of electroporation on outer and inner membrane is different. To clarify
this point, we use the Eq. (2) to calculate the pore densities on these membranes versus
time. We use a trapezoidal pulse with the strength of the electric field is 45×105 V/m, 70
ns of duration and 10 ns of risetime and falltime. The numerical intergrations are taken
with the time-dependent membrane voltages calculated as before.
Obviously, Fig. 5 points out that the value of pore density of inner membrane is
always higher than this of outer membrane. The peak of the inner pore density (about
1019) is proximately 15−16 order larger than that of outer membrane. This point indicates
that ultrashort pulse can excite the inner membrane and the electroporation appears, the
pore current increases dramatically. For this case, the thermal effect can be ignored
because of the short pulses and the low conductivity of biological membranes.
III.2. Power Dissipation with Dielectric Relaxation
We continue to apply our calculation to the normal and cancer cells, interacted with
a triangular pulse with 10 ns rising edge and 50 ns falling edge and the magnitude is 45
kV/cm [2].
216 EFFECTS OF ELECTROPORATION ON BIOLOGICAL MEMBRANES ...
0 5.´10-81.´10-71.5´10-72.´10-72.5´10-73.´10-73.5´10-74.´10-7
Time10-5
0.1
1000
107
1011
1015
1019
Pore Density Hm-2L
Fig. 5. Pore density versus time of a cancer B cell in short pulse case (solid: outer
membrane, dashed: inner membrane)
Firstly, we use the parameters of membranes as in the second and the third columns
of Table. 2. We plot here these two results of transmembrane voltages in two cases.
1.´10-7 2.´10-7 3.´10-7 4.´10-7 5.´10-7 6.´10-7
TIME
1
2
3
4
5
DYOrg, DYCell
Fig. 6. Cell (continuous) and inner (dashed)
voltages of normal B- cell, using typical
parameters.
1.´10-7 2.´10-7 3.´10-7 4.´10-7 5.´10-7 6.´10-7
TIME
1
2
3
4
5
DYOrg, DYCell
Fig. 7. Cell (continuous) and inner (dashed)
voltages of cancer B- cell, using typical
parameters.
We easily see that the voltages of outer membrane always higher than inner mem-
brane in Figs. 6, 7. It is demonstrated that conductivities of transmembranes notably
influence electroporation. With the triangular case, the typical parameters do not make
electroporation take place.
According to the Eq. (2), after a period of time, the pore density reaches No and
therefore, the conductivity of membrane would be higher than in the initial case N = 0.
Therefore, we choose the effective conductivities of the outer and inner normal membranes
according to Joshi’s paper. That means σ1m = 1.1 × 10
−2 and σ2m = 3.1 × 10
−2 as the
initial values. The latter one is chosen as in the very first time of the simulation, the outer
membrane’s conductivity changes dramatically to that value due to electroporation [2]. For
the cancer cells, they are 5.2×10−2 for the outer and 8.6×10−2 for the inner membranes,
respectively.
The only changes in the conductivity give a dramatic difference as in Figs. 8, 9.
With this effective conductivity of outer and inner membranes, the results are in very
NGUYEN THI THUY NHUNG et al. 217
good agreement with Joshi’s results and shows the effects of ultrashort pulses in exciting
sub-stuctures.
1.´10-7 2.´10-7 3.´10-7 4.´10-7 5.´10-7 6.´10-7
TIME
0.5
1.0
1.5
2.0
2.5
3.0
3.5
DYOrg, DYCell
Fig. 8. The voltages of normal B- cell
Dashed: Kotnik’ calculation method (red:
outter, purple: inner), thick: our calculation
method (green: outer, dark: inner), using
updated parameters.
1.´10-7 2.´10-7 3.´10-7 4.´10-7 5.´10-7 6.´10-7
TIME
0.2
0.4
0.6
0.8
1.0
1.2
DYOrg, DYCell
Fig. 9. The voltages of cancer B- cell
Dashed: Kotnik’ calculation method (red:
outter, purple: inner), thick: our calculation
method.(green: outer, dark: inner), using
updated parameters.
In addition, it is well known that a variety of profound biochemical and biophysical
effects can be led by exposure of biological cells to electric field. In common, evaluation
of these effects is based on the power dissipation caused by the exposure. The power
dissipation per init volume is given by P = σE¯2.
Because of the analytical approach, we instantaneously have the distribution of the
potential for the entire space and entire period of pulse time. Therefore, it is convenient
to calculate the dissipation energy of the pulses giving to the cells. Although the power
dissipation is assessed through the bulk properties of the tissue, we ignore all the nonuni-
formities of in the distribution of the field as well as the heat flow that will redistribute
between regions. This is because the exposure takes place in very short exposure time,
10 ns [5]. To compare, we also apply the the infinitive sinusoidal fields towards normal
and cancer cells, Figs. 10, 11. In this calculation, we also include the dielectric relaxation,
an addition component of power dissipation resulted from rotation or flexion of molecular
dipole, which dominants in the higher frequency field, (1012 Hz).
Our calculations point out that the peak values of power dissipation of a cancer cell
and normal cell are somewhat the same (about 1010 W/m3). There is only a frequency
range from 106 toward 108 Hz in which the energy dissipated in the inner membranes
surpasses the external one, but similar for both cases. Excepting this interval, the values
of P of inner membrane of normal cell are higher than those of cancer membrane.
IV. CONCLUSION
To conclude, we so far improve the model of cell membrane exposed to a ns ultra-
short, high-intensity pulse. Basing on the analytical Laplace- transform method of pulses,
we include the effect of dynamic conductivity of cell membranes to achieve better results,
which are in good agreement with numerical calculations. We also concern the differences
218 EFFECTS OF ELECTROPORATION ON BIOLOGICAL MEMBRANES ...
6 8 10 12
LOG10Ω
8
9
10
11
12
LogHPL
Fig. 10. Dissipation power of normal B- cell
versus frequency, (thick: inner mem, solid:
outer mem, dashed: external environment).
6 8 10 12
LOG10Ω
9
10
11
12
LogHPL
Fig. 11. Dissipation power of cancer B- cell
versus frequency, (thick: inner mem, solid:
outer mem, dashed: external environment).
of the responses between cancer and normal B-cells. In detail, the potetials of cancer cells
are less than the potentials of normal cell for a few volts.
Some difficulties, however must be noticed to enhance our results: The time-dependent
conductivity needs to be recreated based on the theory of Krassowska et al. The predica-
ment is that we have to choose the initial pore density without a connection between the
equilibrium density of pores and the effective conductivies of membranes.
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Received 30 September 2009.


Effects of Electroporation on Biological Membranes Exposed to High Potentials

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Effects of Electroporation on Biological Membranes Exposed to High Potentials

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