Biomedical Engineering Reference
In-Depth Information
analytical and computational treatment to theoretically describe the nanoparticle
transport mechanism through an induced topological disorder or an ion pore inside
the cell membrane. A brief outline of this situation is presented below.
We introduce a probability function that describes the trapping probability for
a nanoparticle in a specific ion pore/channel in the membrane. Consider that the
nanoparticles are randomly moving on the surface of the membrane. A nanoparticle
with an average Hamiltonian
H
(Eq.
6.4
) has a non-zero probability for being trapped
inside the
i
th pore as long as the pore cross-sectional area (
A
ch
,
i
(
) is larger than
the nanoparticle diameter (
A
nc
). If the pore ensures that the membrane thickness
vanishes at the core with a cross-section
A
ch
,
i
(
t
)
(see Fig.
6.11
), the trapping rate
will in fact be proportional to the rate of nanoparticle release into the cellular interior
as long as
A
ch
,
i
(
t
)
t
)
≥
A
nc
. The pore cross-section
A
ch
,
i
(
t
)
is considered here to be
time dependent
, due to the fact that the cross-sectional area of the pore is assumed
to change back-and-forth with time within the lifetime (
(
t
)
ms) of the pore/channel,
following the characteristic behavior of the type of pores induced by chemotherapy
drugs (see Fig.
6.10
). However, the novelty found in chemotherapy drug-induced
pores is that
A
ch
,
i
(
∼
)
is usually found to be constant during a channel's lifetime,
following specific energy states in most of the other known channels induced by
antimicrobial peptides.
A
ch
,
i
(
t
)
is proportional to the electrical conductance of the
pore. As the plot of a chemotherapy drug-induced pore conductance versus time is
found to follow a triangular pattern (see Fig.
6.10
),
A
ch
,
i
(
t
should consequently
follow an identical pattern when it is plotted against time within the lifetime of a
channel. Considering all these facts, we propose that the statistical probability (
p
i
(
t
)
)
for a nanoparticle to cross a cell membrane by traveling through a temporarily induced
i
th toroidal pore/channel (see Fig.
6.11
) at zero external field (driving force) is as
follows
t
)
A
ch
,
i
(
t
)
E
np
,
i
k
B
T
p
i
(
t
)
∼
e
for
A
ch
,
i
(
t
)
≥
A
nc
or 0 for
A
ch
,
i
(
t
)<
A
nc
(6.9)
A
cell
Here, the probability is assumed to be proportional to the Boltzmann's function
exp
but it also changes with the changing cross-sectional area of the
specific pore, normalizedwith the total surface area (
A
cell
) of the cell. This probability
is therefore a time-dependent variable. Here,
T
is the absolute temperature and
k
B
is
theBoltzmann's constant.
E
np
,
i
stands for the total binding energy (
U
np-lip
,Eq.
6.5
)of
the nanoparticle inside the pore. Figure
6.12
illustrates this preferential nanoparticle
migration through pores. In the specific case of nonbinding with lipids inside pores,
a nanoparticle experiences a maximum energetic probability to be crossing through a
pore, since for this condition exp
(
E
np
,
i
/
k
B
T
)
approaches the absolute maximum (1).
We suspect that silica nanoparticles and similar ones which exhibit non-interacting
behavior with lipids (as explained in an earlier section) fall within this favorable
nanoparticle category.
If at any time
t
there are
N
ion pores or channels induced across the membrane sur-
rounding a biological cell, the total cell internalization probability for a nanoparticle
p
tot
(
(
E
np
,
i
/
k
B
T
)
t
)
is given by
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