1. Mathematical Process Structuring Cellular Transmembrane Potential Model
M. Stefani
Old Dominion University, Norfolk, Virginia, USA
The focus of this paper is to discuss the various formal mathematical models used in the calcu-
lations for transmembrane potential present in cell walls during exposure to electric fields. This
has direct application in the field of bioelectrical physics where the theoretical effect known as elec-
troporation is caused by these transmembrane potentials. Therefore, in order to structure realistic
simulations these potentials must be accurately calculated using apropos assumptions, approxima-
tions and boundaries. As always, mathematics has supplied the tools to create a useful model, but
the proper tools must be chosen for the right formulations. The distillation of such a model is to be
presented in this paper.
Keywords: Transmembrane, Potenital, Model
I. INTRODUCTION
Within the field of bioelectrical physics, there is a par-
ticular interest in the effects of electric fields on biological
tissue. One such theorized effect is known as electropo-
ration. Electroporation being an increase of permeabil-
ity in a cell membrane cause by a transmembrane po-
tential (TMP). These potentials are induced by pulsed
external electric fields around the cell. The interaction
of these fields with the conductive mediums within and
without the cell, cause a strong electromagnetic interac-
tion in the immediate vicinity of the membrane. Then
through molecular interactions, it causes pores to form
in the bi-lipid layers (the fundamental structures of cell
walls). This effect takes place on a scale that makes direct
visualization all but impossible. We can make molecular
dynamic models for very small sections of a bi-lipid layer
or experiments can be done on full cells, but directly mea-
suring the poration is not currently possible and neither
is complete accuracy in measuring the TMP+
.
Therefore, models to calculate the possible TMP are
extremely important in these studies that have far reach-
ing effects in the field of medicine, biology, and applied
physics. In this paper, different ”generations” of possible
models will be presented and discussed in terms based on
Dr. Adam’s ”idealization of the process of mathematical
modeling” presented in FIG.1
Each generation of model will be an iteration passed
through this feedback loop probing its strengths and
weaknesses in order to build the next model until an ac-
ceptable level of complexity and accuracy is reached.
II. APPROACHING THE PROBLEM
A. General solution to the Laplace Equation
Fundamentally, the desired model will calculate the
potential for any point on the membrane. In order to do
this reasonably, a solution for all points in space is re-
quired. Electrostatically, this means the potential can be
described by a function that solves the Laplace equation.
FIG. 1. Idealization of the process of mathematical modeling
2
Φ = 0 (1)
The Laplace equation has known general solutions for
different symmetries. A uniform electric field applied to
an idealized spherical cell is assumed. The general solu-
tion in this geometry is:
Φ(r, θ, φ) =
∞
l=0
l
m=−l
[Almrl
+ Blm
1
rl+1
]Ylm(θ, φ) (2)
Here, the terms A and B are solved in each region by
using matching conditions(given by potential theory) to
adjoining regions where E0 is the uniform field and σ is
the charge distribution.
lim
r→∞
(− Φ) = E0 (3)
Φi = Φo (4)
ˆn · (σoEo − σiEi) = 0 (5)

2. 2
The Yml terms in the general solution are the Spherical
harmonics. This solution can be simplified because of the
uniform field assumption which would cause an azimuthal
symmetry in the charge distribution on the cell. This
sets the m indices to 0 and this results in the well know
Legendre-polynomials. This gives the ”ready-to-apply”
general solution for the first generation model.
Φ(r, θ) =
∞
l=0
[Almrl
+ Blm
1
rl+1
]Pl(cos(θ)) (6)
III. FIRST GENERATION: SPHERICAL
SYMETRY
As FIG.2 shows, there are permittivity and conductiv-
ity factors to considered inside and outside the cell and
yet another possible value given by the membrane itself.
The first generation model requires some assumptions.
Namely, that the cell wall behaves as a non-conductive
shell and that the material outside and inside the medium
is conductive. Using the general solution, create a system
of equations describing the potential outside and inside
of the cell. Then, solve the system of equations by ap-
plying matching conditions that gives the complete set of
equations that describe the potential throughout space.
Then, solve for the TMP, which is the difference in po-
tential on the outside and inside of the shell.
∆Φ = Φo − Φi (7)
From this, the well established Schwans equation is de-
rived.
∆Φ =
3C
2K
ERcos(θ) (8)
C = λo[3dR2
λi + (3d2
R − d3
)(λm − λi)] (9)
K = R3
(λm+2λo)(λm+
1
2
λi)−(R−d)3
(λo−λm)(λi−λm)
(10)
This is the first generation solution to the potential.
R is the radius of the sphere, d is the thickness of the
cell wall and the λ terms are the various permittivitys.
This rather long expression tells us the potential will be
cos(θ) dependent and directly related to the strength of
the applied electric field.
A. ”Simple as posible, but not simpler”
This is good, but the solution is cumbersome and can
be simplified. According to Kotnik at el.(2000) when
FIG. 2. spherical model
applying the non-conductive assumption λm = 0, as well
as other physical values for λ, this expression C
K is very
accurately approximated to 1. This gives the final first
generation model.
∆Φ =
3
2
ERcos(θ)ˆr (11)
Using the magnitude of EF this calculates the TMP per-
pendicular to the surface with a θ dependence.
B. Errors, assumptions, changes: something is
wrong here, an ellipsoid belongs here
When using all physical values in C and K it changes
the first generation estimate about 0.1%. Meaning, these
details can be safely discarded in this leading term. How-
ever, one assumption made could cause a large difference
in the potential.
Cells aren’t actually perfectly spherical. Due to this,
most other researchers have advanced their models by
adapting to a spheroid. Following a path not traveled,
and perusing a better geometry for a more physically
accurate and generalizable model, this paper will use an
ellipsoid geometry.
IV. SECOND GENERATION: SPHERE TODAY,
GONE TOMORROW
Progressing to an ellipsoid has a much uglier solution.
This is because there is no general solution to the Laplace
equation for this geometry. Therefore, method used to
calculate the potential must adjusted. In order to solve
for an arbitrary geometry, the integrable form of the
laplace equation is required. Starting in ellipsoidal co-
ordinates, the final itegrable equation (Eq.14) is derived.
2
Φ =
4 ϕ(ξ)
(ξ − µ)(ξ − ν)
∂
∂ξ
ϕ(ξ)
∂Φ
∂ξ
= 0 (12)
ϕ(ξ)
∂Φ
∂ξ
= −
E
2
(13)

3. 3
converting ξ into surface element gives
Φ(ξ) =
∞
ξ
Eds
2 ϕ(s)
(14)
where
ϕ(s) = (a2
µ2
+ s)(b2
µ2
+ s)(c2
µ2
+ s) (15)
ξ = 0 on the surface of any scaled ellipse by a factor µ
described by
x2
a2
+
y2
b2
+
z2
c2
= µ2
(16)
From Eq.14 ”shells” of ellipsoidal shapes can be inte-
grated throughout space. Using this fundamental inte-
gral form of potential, the calculation for external poten-
tial can be integrated at µ = 1 and, from there, an inter-
nal potential can be calculated by integrating the scaling
factor µ over the thickness of the cell wall. The differ-
ence of these potentials, along with appropriate matching
conditions, will give the new TMP. Unfortunately, this is
were the model leaves the realm of neat analytic solu-
tions.
These integrals result in a function known as the el-
liptic integral of the first kind and is heavily dependent
on the magnitudes of the axis of the ellipsoid. However,
an approximate answer can be given by an expansion of
this function(Gradshteyn2007) and through this process
the potential for an ellipsoid is solved. The less than
attractive result is given by:
∆Φ = (E·ˆn)
a2
+ b2
+ c2
2
√
a2 − b2
x2
a4
+
y2
b4
+
z2
c4
F(a, b, c) (17)
F(a, b, c) sin−1
√
a2 − b2
a
+
a2
−c2
a2−b2 sin−1
(
√
a2−b2
a )3
6
+...
(18)
So stands the second generation model for TMP per-
pendicular to the ellipsoid surface.
There are several important considerations to make
when analyzing this model. One such consideration is
that this solution is true in the limit where x, y, z satis-
fying the elliptic equation describing the cell wall.
x2
a2
+
y2
b2
+
z2
c2
= 1 (19)
Another important note is that this model is dependent
on orientation of the EF. Its directional normal vector of
the EF will result in different final expressions when con-
verted to ellipsoidal coordinates. It is very pleasing to
note, that when a = b = c a sphere is restored and con-
verting the normal directional vector from the unifrom
EF to radial normal vector resurrects the Schwans equa-
tion (see appendix).
A. Errors, assumptions, changes: can’t have a
perfect model for an imperfect world
Geometrically, and physically this is a good model for
TMP in a constant electric field (EF). However, in the in-
troduction, it is mentioned that the poration is produced
by a pulsed EF*. These pulses can take many forms: Si-
nusoidal, Retangular, Triangular, ect. Our current model
assumes a uniform and constant EF. Physically, as the
EF intensity changed with these pulses, the TMP would
also change. It is possible to make a rough model of these
changes by basing a model on a rectangular pulse. Es-
sentially, this models a cell exposed to a direct current
EF that is suddenly turned off. We assume there will
only be rectangular pulses because other pulses require
unique solutions given by the Laplace transformation†
.
Furthermore, the assumption that after the pulse, the
cell behaves much like a capacitor is made. Capacitors
in this situation will discharge its voltage exponentially.
The next model will tackle this complication.
V. THIRD GENERATION: THERE’S STILL
TIME
The cell is expected to behave as a capacitor that would
discharge exponentially. This kind of discharge can be
modeled by:
∆Φ(t) = ∆Φ[1 − exp(−
t
τ
)] (20)
Here, t is time and τ is the time constant of a capac-
itor. This constant can be derived from the physics of
capacitance in electrostatics.
τ =
R m(2λo + λi)
2dλoλi
(21)
m
d is the capacitance of the membrane. Combining
this multiplicative decaying factor with the second model
produces the final model. Do to the excessive length of
this expression, its explicit representation is omitted.
A. Errors, assumptions, changes: know when to
hold ’em, know when to fold ’em
The remaining errors of this model are extremely com-
plex to resolve. For example, a real world cell would not
have a uniform cell wall. To resolve this, finite element
analysis is required (no thank you). Another example is
the models limitation to a rectangular pulse. This can be
resolved by reducing the model back to a sphere (losing
the greatest strength of this model) and utilizing Laplace
transformation to account for the time varying electric
field.

4. 4
VI. DISCUSSION
Throughout the modeling process good approxima-
tions were made both model and method were scraped
between generations, as a result, a new realistic and pow-
erful model of transmembrane potential is created. The
strength of this new model is based in the more gener-
alized ellipsoid geometry when compared to past mod-
els based on spheroids. By varying the axis length, a
spheres, spheroid and even close approximations to rod-
shaped cells common in bacteria, can be modeled. Such a
model was not created before. The only limitation of this
model is the assumption of rectangular pulse. The next
step for this research is to create a program that will be
able to numerically calculate and graph these potentials,
given cell and experimental parameters.
VII. ACKNOWLEDGMENTS
Many thanks to maths, physics, and those who’ve
taught me.
VIII. APPENDIX: A VESTIGIAL ORGAN,
ONLY OF INTEREST IF THERE’S A PROBLEM
A. Further maths
• Ellipsoidal coordinates:
x2
=
(a2
+ ξ)(a2
+ µ)(a2
+ ν)
(a2 − b2)(a2 − c2)
(A-1)
y2
=
(b2
+ ξ)(b2
+ µ)(b2
+ ν)
(b2 − a2)(b2 − c2)
(A-2)
z2
=
(c2
+ ξ)(c2
+ µ)(c2
+ ν)
(c2 − a2)(c2 − b2)
(A-3)
• (ξ, µ, ν) orthogonal basis:
∂x
∂ξ
∂x
∂µ
+
∂y
∂ξ
∂y
∂µ
+
∂z
∂ξ
∂z
∂µ
= 0 (A-4)
∂x
∂µ
∂x
∂ν
+
∂y
∂µ
∂y
∂ν
+
∂z
∂µ
∂z
∂ν
= 0 (A-5)
∂x
∂ξ
∂x
∂ν
+
∂y
∂ξ
∂y
∂ν
+
∂z
∂ξ
∂z
∂ν
= 0 (A-6)
• Elliptical normal vector conversion:
ˆn =
x
a2 , y
b2 , z
c2
x2
a4 + y2
b4 + z2
c4
(A-7)
• Calculation of integral:
t = s
µ2
∆Φ =
∞
ξ
Eds
2 ϕ(s)
−
1
µ0
µ
∞
ξ
Edt
2 ϕ(t)
dµ (A-8)
where ξ is the greatest root of the function
f(µ, ξ) =
x2
a2µ2 + ξ
+
y2
b2µ2 + ξ
+
z2
c2µ2 + ξ
− 1 (A-9)
and
1 − µ0 = ∆µ = d
x2
a4
+
y2
b4
+
z2
c4
(A-10)
using this and factor P for matching conditions
yeilds
∆Φ = P
E
4
d
x2
a4
+
y2
b4
+
z2
c4
∞
ξ
ds
ϕ(s)
(A-11)
the indefinite elliptic integral is a special function,
elliptic integral of the first kind E.I.
ds
ϕ(s)
=
2E.I. sin−1
√
a2−b2
√
a2−s2
|a2
−c2
a2−b2
√
a2 − b2
+ const.
(A-12)
which expands as
E.I.[β|δ] = β +
δβ3
6
+
(−4δ + 9δ2
)β5
120
+ O[β7
] (A-13)
all of these elements put together, give Eq.17 and
Eq.18 in the paper.
• Ellipse returns to sphere:
a = b = c = r (A-14)
x2
+ y2
+ z2
= r2
(A-15)
expand sin−1
sin−1
√
a2 − b2
a
=
√
a2 − b2
a
+ ... (A-16)
choose E to be uniform in ˆz
ˆz · ˆn = rcos(θ) · ˆr (A-17)
substituting all of these new conditions
∆Φ = (Ercos(θ)ˆr)
3r2
2
√
a2 − b2
x2 + y2 + z2
r4
√
a2 − b2
r
+ 0 + ...
(A-18)
which simplifies to Schwan’s equation
∆Φ =
3
2
Ercos(θ)ˆr (A-19)

5. 5
B. Footnotes
* The EF must be pulsed because in order for poration
to occur. This is because the field strength must be very
large and if left constant the cell would simply rip apart.
† This is reasonable after reading Kotnik 1998 paper.
Rectangular pulses are also the most commonly used
pulse form.
+ Transmembrane potentials translate to physical
voltage across the membrane.
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