Download The surface charge of a cell lipid membrane

The surface charge of a cell lipid membrane
M. Pekker 1 and M. N. Shneider 2
1
MMSolution, 6808 Walker Str., Philadelphia, PA 19135, USA
2
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544,
USA
E-mail: [email protected] and [email protected]
Abstract
In this paper the problem of surface charge of the lipid membrane immersed in the physiological solution is
considered. It is shown that both side of the bilayer phospholipid membrane surface are negatively charged.
A self-consistent model of the potential in solution is developed, and a stationary charge density on the
membrane surface is found. It is shown that the ions of the surface charge are in a relatively deep (as
compared to kBT) potential wells, which are localized near the dipole heads of phospholipid membrane. It
makes impossible for ions to slip along the membrane surface. Simple experiments for verifying the
correctness of the considered model are proposed. A developed approach can be used for estimations of the
surface charges on the outer and inner membrane of the cell.
Introduction
The question about the distribution of electric charge near the cell membranes is the key for many problems
associated with the interaction of cells with external electromagnetic fields [1-6]. For example, if ions are not
electrically neutral, the areas are not strongly bound with the membrane and can move freely along it, the
electromagnetic field has a weak influence on the membrane. If the ions are firmly bound to the membrane,
the electrical component of electromagnetic fields could lead to the membrane deformation [7, 8].
The surface charge of the cell membrane has been studied in many experimental works. We will not consider
all works and may refer only to [9,10], where the experimental data of the surface charge of cell membranes
obtained by different methods are given. A range of results is broad enough: σ m = 0.3 − 0.002C / m2 .
The problem of spatial distribution of charge in the vicinity of the biological membranes surface has been
considered in many теоретических papers, see eg [1-5]. In all these works, the near-surface potential of the
membrane was considered under the Gouy-Chapman theory [11, 12] or its later modification by Stern [13], in
which the charge on the membrane surfaces is considered to be given. In these theories, the membrane was
considered as a continuous dielectric, without taking into account its fine structure, and a surface charge was
determined on the basis of the electrochemical properties of the dielectric surface (see, eg [14, 15]).
In the classical Gouy-Chapman-Stern theory the surface charge ions can freely slide along the surface.
Because, it is assumed in this theory, that the interaction of ions with the surface occurs due to electrostatic
forces witth the induceed charges and absorption forces, whichh depends onnly on the disstance of the ion to the
surface. In
n the case of a phospholipiid membrane,, the electrolyyte ions interaact with phosppholipid dipooles, which
form a mo
osaic (lattice)) structure (F
Fig.1a) [16]. As
A a result, thhe surface chharge ions aree in a periodicc potential
(Ris.1b), i.e.
i sliding theem along the surface of the membrane is suppressedd. This fact is crucial in thee theory of
interaction
ns of the weaak microwave field with the
t cell mem
mbranes, propoosed in [17], where it was assumed
that the lo
ongitudinal electric field of
o the microw
wave acting oon the surface charge cauuses forced loongitudinal
vibrationss of the memb
brane, because the surface charge ions aare rigidly bonnded to the suurface and cannnot slip.
Fig. 1a. A simplified mosaic modeel of a phosph
holipid membbrane [16]. Sppheres represeent the dipolee heads on
the inside and the outsiide of the mem
mbrane.
Fig. 1b. A fragment of a two-dim
mensional disstribution patttern of potenntial along m
membrane surrface, at a
distance equal
e
to 0.15 of
o the lattice period
p
(a desccription of thee computationnal model is ggiven in Partss 1 and 2).
In our pap
per, we take into
i
account the fine struccture of the biilayer phosphholipid membbranes [16]. It is shown
that the su
urface of the cell
c membran
ne is charged negatively w
with ions, trappped in the pootential wells formed by
the dipolee heads of thee membrane phospholipids
p
s. These ions are strongly bbound to the membrane suurface: the
ion bindin
ng energy U with the meembrane surfface is much greater thann the thermal energy kBT. That fact
allowed the
t constructiion of a selff-consistent th
heory of Gouuy-Chapman and Stern thheories, and helped to
determinee the average charge densitty of sitting on the membraane.
The mem
mbrane surfacce charge theeory presented
d in this papper supports the idea [17] of electrom
mechanical
impact off weak microw
wave electrom
magnetic field
ds on the surfface density rredistributionn of transmem
mbrane ion
channels.
We emphasize that this article does not discuss the passage of ions through the transmembrane ion channels or
pores. We restrict ourselves to considering only the surface processes in the lipid membrane placed in the saltwater solution.
The main ions in the physiological solution аrе: Na+, K+, Са2+, Mg2+ and Сl-. The total concentration of these:
ni ≈ 1.8 ⋅ 10 26 m-3 (300mmL/Mole) [21], но для простоты мы рассмотрим только случай, когда the
bilayer phospholipid membranes immersed in water solution of NaCl.
1. A model of the phospholipid membranes.
The dielectric permittivities of the membrane and water are correspondingly:
εm ≈ 2 and ε w ≈ 80 [18, 19]. It
would seem that in order to evaluate the force acting on the ion of charge q near the membrane surface the
formula [20] can be applied:
q2 ε w − ε m
Fq = 2
4h ε 0 (ε w + ε m )
(1)
Here ε0 is the permittivity of free space, and h is the distance to the interface between dielectrics. However,
the paradoxical result follows from the formula (1) that the ion cannot get from the dielectric with the higher
dielectric permittivity into the dielectric with the lower dielectric permittivity, because the force acting on it
increases inversely proportional to the square of the distance to the boundary between dielectrics. This
statement contradicts the experimental facts and theoretical models of Hodgkin and Huxley’s kind, based on
the phospholipid membrane permeability to ions. Usually, in theories of the interaction of ions with the
surface of dielectrics, the minimum value of h is selected as the size of the ion, or the Debye radius, if the
insulator’s surface is saturated with ions [15]. However, this does not resolve the problem of phospholipid
membranes "impenetrability". In fact, the problem lies in that the formula (1) is derived in the approximation
of continuous medium with the dipoles of the infinitely small size, without taking into account the real
structure of the surface layer of the membrane.
It is known that the phospholipid molecules of cell membrane are forming a mosaic (matrix) structure in
which dipole heads are directed towards the liquid (positively charged head faces outward membrane) [16].
The average surface area per molecule of the lipid is ≈ 0.5 nm2, the length of the polar head is ~ 0.5 −1nm,
the radius of the head is ~ 0.2 − 0.3 nm, and the distance between the hydrophilic heads of the membrane is
in the range of 5-7 nm [21, 22]. The dipole moment of the phospholipid head is 18.5-25 D [23] (1 D
=3.34·10−30 C.m), i.e. more than 10 times greater than the dipole moment of water molecules. On the basis of
geometrical dimensions of the cell membrane and the size of water molecules, it can be concluded that the free
space between the head does not exceed the size of a water molecule (~ 0.2nm). That is, the membrane,
interacting with the ions of surrounding liquid, cannot be considered as a dielectric medium with an
infinitesimal dipoles size.
These facts allow the consideration of the following simplified model of the ion interaction with the
membrane:
1. The membrane represents a matrix (Fig. 2) with a mesh size a × a . In the nodes of cells the dipoles are
located; the dipole charge is q; the distance between the charges (the dipole length), d; the distance
between the dipoles along the axis z, l.
2. An ion is a classical particle and cannot approach a dipole at a distance less than the sum of the radii of the
head and the size of the ion. It is important that the ion can approach the membrane dipole heads close
enoug
gh that would
d be "captured
d" by the poteential well. T
This is a standdard assumptiion in the theeory of the
interaaction of ions with the surfa
face of the dieelectric [15]. S
Since the dipoole moment oof water is 10 times less
than the
t dipole mo
oment of the phospholipid
d head molecuule and near the head can not be more than onetwo water
w
moleculles, the interaaction betweeen the ions loocated near thhe surface off the membranne and the
waterr molecules can
c be negleccted, as comp
pared with th e interaction of the ions w
with the dipooles of the
memb
brane.
3. We want
w
to emph
hasize one mo
ore time that ionic permeaability membbrane is a sepparate problem
m and not
consid
der in this wo
ork, for us it iss important to
o know how sstrong ions atttached to surfface of membbrane.
y
a
a
x
m
modell of a membraane: the mem
mbrane is a m
matrix with dippoles in the nnodes. a is
Fig. 2. A simplified mosaic
the mesh size of the matrix,
m
d is thee distance between the chaarges of the diipole head, l is the distancce between
the dipolees along the z axis.
The main ions in the physiological
p
solution аrе: Na+, K+, Са2 +, Mg2+ and С
Сl-. The total concentrationn of these:
ni ≈ 1.8 ⋅10 26 m-3 (3000mmL/Mole) [21], but for
f simplicityy we consideer only the case when thhe bilayer
phospholiipid membran
nes are immerrsed into the water
w
solutionn of NaCl.
otential near the surface of
o the membrrane
2. The po
The expreession for thee membrane potential in a cylindrical coordinate syystem
surface off the membran
ne (Fig. 3):
(r, θ , z) at a point above the
U (r , θ , z ) =
n
n
∑ ∑ U (r ,θ , z ) =
i =− n j =− n
i, j
q ⎛⎜
1
∑
∑
⎜
2
2
2
i = − n j = − n 4πε 0 ⎜
⎝ r + ri , j − 2r ⋅ ri , j cos(θ − θ i , j ) + z
n
n
−
−
+
1
r + r − 2r ⋅ ri , j cos(θ − θ i , j ) + ( z + d )
2
2
i, j
(2)
2
1
r 2 + ri 2,j − 2r ⋅ ri , j cos(θ − θ i , j ) + ( z + l )
2
1
r 2 + ri 2,j − 2r ⋅ ri , j cos(θ − θ i , j ) + ( z + l + d )
2
⎞
⎟
⎟
⎟
⎠
Here, the radial and angular coordinates ri, j , θ i, j correspond to the position of the dipole in the node with the
numbers (i, j), q is the dipole charge, and ε 0 is permittivity of free space. The value of n in (2) is chosen such
that the potential near the membrane does not depend on the size of the matrix n. The count goes from the
(
node (dipole head) (Fig. 3): r = x 2 + y 2
)
1/ 2
, θ = arctan( y / x ) .
We expand the potential U in a Fourier series by θ . Taking into account the rotational symmetry on
rotations for the accepted square matrix of the dipole heads, we obtain:
J
J
j =1
j =1
U (r , θ , z ) = U 0 (r , z ) + ∑ U j (r , z ) cos(4 jθ ) + ∑ V p (r , z ) sin (4 jθ )
U 0 (r , z ) =
2
π /2
∫ U (r , θ , z )dθ
π
U j ≠ 0 (r , z ) =
V0 (r , z ) = 0
(3)
0
4
π
π /2
∫ U (r , θ , z ) cos(4 jθ )dθ
0
V j ≠ 0 (r , z ) =
4
π
π /2
∫ U (r , θ , z ) sin (4 jθ )dθ
0
π /2
Fig.3. Spaatial lattice of dipoles in bilayer
b
lipid membrane.
m
Pootential is callculated at thhe point P. Ellipsoids at
lattice sitees correspond
d to the dipolee heads of pho
ospholipid moolecules.
In the fram
mework of th
he model preseented above, estimates shoow that the innfluence of thee dipoles locaated on the
lower sidee of membran
ne (Fig. 2) on the potential distribution U at z > 0 ccan be neglectted. The samee is true to
the effectt of dipoles in the upperr side of thee membrane on the potenntial at z < −(l + d ) . As far as the
dependence of U on θ , in the region r ≤ a / 2 = 0.35 nm thhe contributioon of above thhe zero harmoonics does
not exceed
d 7%, so we will
w hencefortth neglect theem.
For examp
ple, let consid
der the case, when bilayerr phospholipidd membrane iis immersed iin the NaCl ssolution, at
−19
the follow
wing set of parrameters: a = 0.7 nm, d = 0.5 nm, l = 8 nm, and q = 1.6 ⋅ 10
C [16, 21].
Fig. 4. sh
hows contour plot of reducced U / kBT for a singly ccharged ion aat T = 300K, Fig. 5 the deependence
U / kBT on
o the axis z.
1.1
100
90
1
80
1
0.9
1
70
1
0.8
z/a
60
0.7
5
50
5
0.6
40
10
5
0.5
30
10
20
20
30
20
30
40
0.3
80
10 0
0
0.05
0.1
60
0.15
40
30
0.2
20
0.25
10
10
60
5
0.4
0.3
0
r/a
Fig. 4. Contour plot U / kBT , corresponding to a half of the cell shown in Fig. 1b.
100
r/a=0
r/a=0.14
r/a=0.21
r/a=0.2975
4
U/kBT
10
1
2
3
1
0.1
0.25
0.50
0.75
1.00
z/a (a=0.7nm)
Fig. 5. The parameter
U / kBT dependence on z. Curve 1 corresponds to r / a = 0, 2 – r / a = 0.14, 3 - r / a =
0.21, and 4 – r / a =0.30.
It is seen from Fig, 5 that the value
U / kBT decreases exponentially with the distance from the membrane,
wherein a distance z / a = 0.5 , the ratio
U / kBT is about 10. That is still sufficiently deep potential well, but
already at z / a = 1 , the influence of the membrane on the ions can be neglected.
An effective potential of the ion across the membrane can be determined. Let the ion be a sphere of radius ri ,
and rh is the distance from the positive charge of the dipole to the head of the ion (Fig. 6).
z
ri
rh
Ion
Dipole head
Fig. 6 The negative ion located near the head of the phospholipid molecule dipole. Ion is a charged sphere of
radius ri , and rh is the distance from the positive charge of the dipole to the head of the ion.
In this case, the effective reduced ion potential,
Uef
2π 3
=
kBT kBT 4πri3
ri2 − ( rh + ri − z )2
rh + 2 ri
∫
rh
dz
∫U (r, z )rdrdz
(4)
0
The dependence of the effective potential of the ion Cl − in the potential well on the size of the head rh is
shown in Figure 6. The chlorine ion radius ri = 0.18 nm, and the atomic weight is M i = 35.4 .
It is seen that for the typical size of the dipole head rh = 0.2 nm ( rh / a = 0.29 ) and the ion radius ri = 0.18
nm, the reduced potential depth is U ef / k B T ≈ 10 . Thus, the trapped ion is quite firmly bound to the
membrane. It should be noted that for electrolytes of a different composition of negative ions, for example in
the axon, where the major negative ions are anion groups of macromolecules and phosphates, estimations of
the potential will differ from that shown in Fig.7.
35
30
U/kTB
25
20
15
10
5
0
0.15
0.20
0.25
0.30
0.35
5
0.40
0.4
45
0.50
rh/a (a=0.5n
nm)
Fig. 7. Th
he dependencee of the effective potentiall of the ion onn the size of th
the head at thee assumed raddius of the
ion ri = 0..18 nm (Cl-).
3. Evalua
ation of the membrane
m
su
urface chargee.
For simplicity, we
w assume thee ions as a point charges, bbut will take iinto account tthe finite sizee of the ion
as the minimal distancce that the io
on can approaach to the m
membrane. As indicated abbove, this is a standard
he dielectric ssurfaces.
approach in models of the ions interraction with th
harge on the membrane
m
dooes not affectt its structuree, i.e. the dippole heads.
We assume that the space ch
m
(w
without taking
g into accountt the ionic chaarge on it) att a distance off the order
Since the field of the membrane
s is negligiibly small, an
nd the distancee between thee free ions inn the solution is larger thann the mesh
of mesh size
size, we can
c relate the charge on the membrane surface to thee charges in ssolution by eqquating the fllows at the
membranee/liquid boun
ndary (Fig. 8)..
Fig. 8. A scheme of th
he ions flows near the mem
mbrane. The ions flux to tthe membranne and “evapooration” of
ntial wells of the
t depth U.
the ions frrom the poten
Let us esttimate the flux
x of ions falliing on the meembrane. Assuuming the Booltzmann disttribution of ioons outside
the memb
brane:
⎛ qϕ ⎞
⎟⎟ ,
n = n∞ exp⎜⎜ −
k
T
⎝ B ⎠
where
(5)
n∞ - is ion density at infinity. Assuming that the ions are in thermal equilibrium with the water
molecules, we obtain the following estimate for the ion flux on the surface of the membrane:
1/ 2
⎛ M H 2O ⎞
⎟⎟
Pin =
n∞vH 2O ⎜⎜
2 π
⎝ Mi ⎠
1
⎛ M H 2O
Here v H 2O ⎜⎜
⎝ Mi
⎞
⎟
⎟
⎠
⎛ qϕ ⎞
⎟⎟ .
exp⎜⎜ −
⎝ kBT ⎠
(6)
1/ 2
is the averaged ion velocities in solution, and M H 2 O , M i are the masses of the water
molecule and the ion, correspondingly.
On the other hand, an estimate of the ion flux “evaporating” from the surface of the membrane is as follows.
The flow of ions from the surface of the membrane can be considered as a process of evaporation of the liquid
with a work function equal to the potential well U. The ion in the well is exposed to random impacts by water
molecules, while the ion’s presence time in the potential well τ i can be estimated as:
⎛ Mi ⎞
⎟(exp(U / kT ) − 1) .
τ i ≈ τ i,H 0 ⎜
2
(7)
⎜ MH O ⎟
2
⎠
⎝
(
Here τ i , H 2 0 M i / M H 2O
)
is the characteristic time of energy exchange between the ion and the water
molecules. In (7) we have taken into account that at U / kT << 1 the ions are reflected from the membrane,
i.e., the time delay is zero. For estimates τ i , H 2 0 can be assumed equal to the collision time of water molecules
−9
in a solution τ i , H 2 0 = λH 2 O / vT , λH 2 O = 0.2 ⋅ 10 nm is the mean free path of a molecule of water, and
vT is
the thermal velocity of water molecules. Accordingly, the number of the ions leaving the unit surface of the
membrane per unit time is equal to:
Pout =
N i vH 2 0 ⎛ M H 2 O ⎞
1 Ni
1
⎜⎜
⎟⎟
.
=
2
2
a τi
a ⋅ λH 2 O ⎝ M i ⎠ exp (U / k BT ) − 1
(8)
Here 1 / a 2 is the number of potential wells per unit membrane surface; N i is the relative population of the
potential wells with ions. Equating the incident flux of the ions on the membrane to the "evaporating" flow
from it, we obtain the relative population of the potential wells:
1/ 2
⎛ Mi ⎞
1
Ni
⎟
=
n∞ a 2 ⋅ λH 2 O ⎜
⎜ MH O ⎟
1 − Ni 2 π
2
⎝
⎠
⎛ eϕ ⎞⎛
⎛ U ⎞ ⎞
⎟⎟ − 1⎟ .
exp⎜⎜ − 0 ⎟⎟⎜⎜ exp⎜⎜
⎟
⎝ k BT ⎠⎝
⎝ k BT ⎠ ⎠
(9)
In (8) we took into account that the probability of the incident ion being captured for in a potential well is
proportional to 1 − Ni .
Now we will find the distribution of electric potential in the liquid. Following the Debye approximation for
the liquid electrolyte [11]:
ϕ = ϕ 0 exp(− x / λ D )
Where λ D =
εε 0 k B T
2n∞ q 2
(10)
is the Debye length.
The field on the membrane surface:
∂ϕ
∂x
=
x =0
σ
2εε 0
(11)
Where σ is the surface charge density. For the case when the upper and lower surfaces of the membrane are
equally charged σ = 2 N i q / a 2 , we obtain the potential at the membrane surface:
ϕ0 =
N
λD q ⋅ N i
k BT
=
⋅ 2i
2
εε 0 a
2εε 0 n∞ a
(12)
Substituting (12) into (9), we obtain the equation for the value of N i :
1/ 2
⎛ Mi ⎞
Ni
1
⎟
=
n∞ a 2 ⋅ λH 2 O ⎜
⎜ MH O ⎟
1 − Ni 2 π
2
⎝
⎠
η=
q
a2
⎛
⎛ U ⎞ ⎞
⎟⎟ − 1⎟
exp(− ηN i ) ⋅ ⎜⎜ exp⎜⎜
⎟
k
T
⎝ B ⎠ ⎠
⎝
(13)
1
2εε 0 n∞ k BT
2
−18
For parameter values that are typical for a physiological solution: n∞ = 0.9 ⋅10 m-3, a = 0.5 ⋅ 10 m2,
26
T = 300K , λ H O = 0.2 nm, for the chloride ion (M i / M H 2O )1 / 2 ≈ 1.41we obtain from (13) η ≈ 13.9 , and,
2
therefore,
Ni
= 0.025 ⋅ (exp(U / kBT ) − 1) ⋅ exp(− 13.9 ⋅ Ni )
1 − Ni
The calculated dependence of N i on the reduced potential
(14)
U
is shown in Figure 9.
k BT
0.32
1
2
0.28
3
2
σ=[C/m ] ~ qNi
0.24
0.20
0.16
4
0.12
5
0.08
0.04
6
0.00
0
2
4
6
8
10
12
14
16
18
20
22
24
U/kBT
Fig. 9. Dependencies of the relative occupancy of the membrane N i and the surface charge density σ on the
reduced potential
U / kBT . Curve 1 corresponds to n∞ = 5⋅10 26 , 2 – n∞ = 1026 , 3 – n∞ = 5⋅1025 , 4 –
n∞ = 10 25 , 5 – n∞ = 5⋅10 24 , 6 – n∞ = 1024 .
The maximum surface charge density in Fig. 9 corresponds to the case where all of the cells (Fig. 1b) are full,
i.e. one negative ion is in every cell. It is seen that, even for small values U / kBT , the relative occupancy of
the potential wells with charges is large enough and the charge density of the charges, which are tightly
bounded to the membrane, can reach hundredths of coulomb per square meter. It should be noted that in this
estimation we did not consider the interaction between the ions bonded to the membrane, which can certainly
play an important role for a more accurate calculation of the occupancy of the potential wells.
Let us estimate the charge of the free ions of one sign in the non-quasineutral region in the vicinity of the
membrane surface.
q free ≈ q ⋅ n∞ ⋅ rD ≈ 0.013 C / m 2
For typical values of the potential
(15)
U / kBT ≈ 5 − 15, which correspond to the size of the head rh ~ 0.2 − 0.3
nm (Fig.9), a bounded charge sitting on the membrane surface is much larger than the charge associated with
unbounded ions.
4. Discussion
In this article, we examined the behavior of the membrane surface with the dipole heads placed in an
electrolyte solution. However, despite of this very simple approach, the results are directly related to the
processes in biological lipid membranes, such as the membranes of the axon. Because the negative ions inside
of the axon differ from those located on the outside (inside the main negative ions are anion groups of
macromolecules and phosphates, but chloride ions outside), it is expected that the surface charge density of
ions sitting on the membrane is different. Figure 10 shows the potential distribution inside and outside of the
membranee where the inner and ou
uter sides of the membranne have the ssame (A) andd different (B
B) surface
charges.
Fig. 10. Potential disstribution inside and outsside the mem
mbrane whenn the inner aand outer siddes of the
membranee have the sam
me (A) and diifferent (B) su
urface chargees.
A few wo
ords about th
he experimen
ntal verification of the prooposed modeel. As follow
ws from (13),, the main
parameterrs that determ
mine the relattive occupanccy of the mem
mbrane with negative ions Ni , are η ∝ 1 / N ∞
and U / k BT . Accordin
ngly, by chan
nging the den
nsity of NaCl in the solutioon and measuuring the surfaace charge
of the meembrane, the curves,
c
similaar to those sh
hown in Fig.88, can be reprroduced. It is shown in [17] that the
longitudin
nal displacem
ments of the membrane
m
in the microwavve field is resonant and iss linearly deppendent on
the surfacce charge den
nsity (Fig. 10). Accordingly, by channging the saliinity of the w
water, it is ppossible to
determinee the surface charge
c
of the membrane an
nd to comparee with estimaates obtained aabove.
In [24,25
5], the distrib
bution of ion
ns near the surface
s
of a phospholipid membranee was calculaated using
molecularr dynamics, however,
h
in th
hose articles the binding energy of thee ions with pphospholipid hheads was
not illustrrated. Thereffore, it would
d be interestiing to compaare the corresponding ressults of our m
model and
calculations performed
d by using thee molecular dy
ynamics.
Conclusio
ons
It is show
wn in this papeer that:
1. The
T effective binding
b
energ
gy of the ion with
w the mem
mbrane is of oorder of severral kBT. i.e., thhe ions are
fiirmly bounded
d to the surfacce of the mem
mbrane.
2. The
T value of th
he potential decays
d
exponeentially with ddistance from
m the membraane, so that at a distance
grreater than 0.7 nm the influ
uence on the ions of the ellectrolyte by tthe membranee can be negleected.
3. The
T surface ch
harge ions aree in relatively
y deep potenttial wells, loccalized near tthe dipole heeads of the
ph
hospholipid membrane,
m
wh
hich preventss them from slliding along tthe surface off the membranne.
4. A qualitative self-consisten
s
nt theory of th
he potential diistribution neear the membrrane is considdered. It is
sh
hown that thee density of th
he bound charrges on the meembrane can reach hundreedths of C/m2.
5. This
T work sup
pports the assumption that the ions locaated on the eexcitable axonn membrane are tightly
bo
ound with it,, which underlies the worrk [17]. Therrefore, the eleectric componnent of the m
microwave
fiield, interactiing with the ions, transfeers energy annd momentuum directly too the membrrane. This
in
nteraction lead
ds to forced mechanical
m
viibration of thee membrane aand, as a resuult, to a redistrribution of
trransmembranee protein ioniic channels.
It should be
b noted that the proposed
d model of the potential diistribution neaar the membrrane can be exxtended to
for differeent types of biological mem
mbranes, takin
ng into accouunt their charaacteristics.
Fig. 11. Membrane deformations
d
caused by the
t microwavve electric ffield parallel to the surfaace of the
membranee; σ 1 , σ 2 aree the inner an
nd outer surfacce charge dennsities.
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