entropy
Article
Fluctuation-Driven Transport in Biological
Nanopores. A 3D Poisson–Nernst–Planck Study
Marcel Aguilella-Arzo, María Queralt-Martín † , María-Lidón Lopez and Antonio Alcaraz *
Laboratory of Molecular Biophysics, Department of Physics, Universitat Jaume I, Av. Vicent Sos Baynat s/n,
12071 Castellón, Spain; [email protected] (M.A.-A.); [email protected] (M.Q.-M.); [email protected] (M.-L.L.)
* Correspondence: [email protected]; Tel.: +34-964-72-8044
† Current affiliation: Program in Physical Biology, Eunice Kennedy Shriver NICHD, National Institutes of
Health, Bethesda, MD 20892, USA.
Academic Editors: Giancarlo Franzese, Ivan Latella and Miguel Rubi
Received: 28 January 2017; Accepted: 9 March 2017; Published: 14 March 2017
Abstract: Living systems display a variety of situations in which non-equilibrium fluctuations
couple to certain protein functions yielding astonishing results. Here we study the bacterial channel
OmpF under conditions similar to those met in vivo, where acidic resistance mechanisms are known
to yield oscillations in the electric potential across the cell membrane. We use a three-dimensional
structure-based theoretical approach to assess the possibility of obtaining fluctuation-driven transport.
Our calculations show that remarkably high voltages would be necessary to observe the actual
transport of ions against their concentration gradient. The reasons behind this are the mild selectivity
of this bacterial pore and the relatively low efficiencies of the oscillating signals characteristic of
membrane cells (random telegraph noise and thermal noise).
Keywords: non-equilibrium fluctuations; ion transport; biological channel; electrodiffusion;
computational biophysics
1. Introduction
Energy-transduction processes occurring in biosystems display astonishing high efficiencies that
could be partially explained considering the contribution of non-equilibrium fluctuations [1]. Ratchet
mechanisms take advantage of functional anisotropies to bias the response of the system to external
fluctuations and produce a net flux of energy [2]. In the case of nanochannels, asymmetric conduction
has been shown to be a particularly effective ratchet [3–5]. Thus, when the electric voltage across the
membrane fluctuates (due to chemical reactions out of equilibrium or stochastic opening and closing
of membrane channels) [6], a net flow of ions (the so-called stochastic pumping or Brownian pumping)
could appear without the expenditure of metabolic energy due to the current rectification [3,7].
In a recent study, we showed that OmpF porin, a non-specific channel located in the outer
membrane of Escherichia coli (E. coli) [8,9], may perform as a molecular ratchet in conditions designed
to mimic acidic stress on bacteria [7]. Using asymmetrically charged lipid bilayers and realistic pH
gradients we demonstrated that zero-average electric potentials similar to those actually measured in
E. coli [10] may yield electrical pumping of ions against an external concentration gradient. Interestingly,
this uphill transport obtained in the OmpF channel is directional: depending on the orientation of the
concentration gradient, the system can pump either cations or anions from the diluted solution to the
concentrated one [7].
The theoretical study of fluctuation-driven transport in OmpF is complicated by the fact that we
deal with a wide mesoscopic channel allowing the simultaneous transport of water molecules and
salt cations and anions. Furthermore, at extreme pH conditions, protons or hydroxyls also contribute
themselves to the measured current, changing the permeation pathways of salt ions [11]. Accordingly,
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Entropy 2017, 19, 116
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the relative contribution of each ion to the total current (the channel selectivity) cannot be easily
anticipated because of the coupling between electrostatic and diffusional effects arising from the
different diffusivities of ions [11]. Here we address this question using the three-dimensional (3D)
structure of the channel available at the Protein Data Bank (code 2OMF). We calculate the ion fluxes
through the channel using the three-dimensional Poisson–Nernst–Planck (PNP-3D) model and taking
into account the salt ions (K+ , Cl− ) and also protons and hydroxyls (H+ , OH− ).
In principle, the transport of ions against their concentration gradient is attained when the
superimposed fluctuating potential is high enough to reverse the direction of the current that is
originated by the concentration gradient itself. However, in non-ideally selective channels such as
OmpF the reversal of the current could be due not only to the uphill transport of counterions but
also to the downhill transport of coions [7,12]. By means of the PNP-3D formalism, we evaluate the
individual current carried by each ion under different zero-mean oscillating voltages and discuss the
conditions requested to obtain the actual transport of ions against their concentration gradient. We
underscore the idea that due to the multiionic character of the channel, the actual uphill transport
requires considerably larger potentials than those needed to just reverse the direction of the current.
2. Materials and Methods
2.1. Theoretical Calculations
The so-called Poisson–Nernst–Planck equations (PNP) are mean-field phenomenological
equations that describe ion transport through ionic channels [13–15]. The Poisson equation relates
the position-dependent electric charge density to the electrostatic potential of the system, while the
contribution from the ions in solution is estimated using the equilibrium Boltzmann equation. Finally,
the non-equilibrium ionic fluxes are calculated using the Nernst-Planck equations. The channel fixed
charge is obtained by using the University of Houston Brownian Dynamics (UHBD) code [16,17] to
calculate the apparent pKa of the channel residues at the particular pH configuration of our study,
using the three-dimensional structure of the OmpF channel (Protein Data Bank code: 2OMF) as
obtained from X-ray analysis [18]. A more detailed description of the method employed can be found
in [11], including the implementation of H+ and OH− ions into the system. The numerical solution
of the system equations has been obtained using FiPy [19], a solver of partial differential equations
written in Python [20], as described in detail elsewhere [11]. The existence of a charged membrane
was simulated adding a small charged region, 5 Å wide, at one of the channel sides, over the ion
inaccessible membrane region. The charge was adjusted to mimic a surface charge of about 1 negative
elementary charge every 44.2 Å2 (approximately the area per lipid for dipalmitoyl phosphatidylcholine
(DPPS) lipid [21]).
2.2. Experimental Methods
Wild-type OmpF, kindly provided by Dr. S. M. Bezrukov (NICHD, NIH, Bethesda, MD, USA),
was isolated and purified from an E. coli culture. Planar membranes were formed by the apposition
of monolayers across orifices with diameters of 70–100 µm on a 15-µm-thick Teflon partition using
diphytanoyl phosphatidylcholine (DPhPC) or diphytanoyl phosphatidylserine (DPhPS). The orifices
were pre-treated with a 1% solution of hexadecane in pentane. An electric potential was applied using
Ag/AgCl electrodes in 2 M KCl, 1.5% agarose bridges assembled within standard 250 mL pipette
tips. The potential was defined as positive when it was higher on the side of the protein addition
(the cis side of the membrane chamber), whereas the trans side was set to ground. An Axopatch 200B
amplifier (Molecular Devices, Sunnyvale, CA, USA) in the voltage-clamp mode was used to measure
the current and applied potential. The signal was digitalized at 50 kHz sampling frequency after
10 kHz 8-pole in-line Bessel filtering. The chamber and the head stage were isolated from external
noise sources with a double metal screen (Amuneal Manufacturing Corp., Philadelphia, PA, USA).
Entropy 2017, 19, 116
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The pH was adjusted by adding HCl or KOH and controlled during the experiments with a GLP22 pH
meter (Crison, Barcelona, Spain). Measurements were obtained at T = (23 ± 1.5) ◦ C.
3. Results
and2017,
Discussion
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19, 116
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3.1. The Importance
Membrane
Asymmetry
experimentsofwith
a GLP22 Charge
pH meter
(Crison, Barcelona, Spain). Measurements were obtained at T =
(23 ± 1.5) °C.
The mechanisms by which E. coli can survive inside the stomach under considerable acidic
stress (pH
2~3) are
under debate [22–26]. In particular, the role of the outer membrane and
3. Results
andstill
Discussion
its constituents remains mostly unexplored. The existing literature focuses on how proton influx is
3.1. The Importance of Membrane Charge Asymmetry
reduced by changes in membrane fluidity [10], albeit the possible changes in the membrane charge are
mechanisms
byexperimental
which E. coli can
survive
inside thebefore
stomach
under considerable
acidicof lipid
not examinedThe
in detail.
Recent
studies
mentioned
highlight
the crucial role
stress
(pH
2~3)
are
still
under
debate
[22–26].
In
particular,
the
role
of
the
outer
membrane
and
charges on OmpF channel conductance [7], providing the current rectification required for theitsratchet
constituents remains mostly unexplored. The existing literature focuses on how proton influx is
mechanism. We tackle this issue by considering that because of the acidic stress, the outer membrane
reduced by changes in membrane fluidity [10], albeit the possible changes in the membrane charge
could become
charged.
We represent
this situation
using
negatively
charged
are notasymmetrically
examined in detail.
Recent experimental
studies
mentionedby
before
highlight
the crucial
role lipids
in the inner
sidecharges
kept aton
neutral
andconductance
neutral lipids
the outer
facing an
acidicfor
solution
of lipid
OmpF pH
channel
[7], in
providing
themonolayer
current rectification
required
the ratchet
mechanism.
issueheads
by considering
of the acidic this
stress,
outer
(representing
the titration
ofWe
thetackle
lipidthis
polar
at low that
pH because
[27]). Although
isthe
a convenient
become
charged. interactions,
We represent this
by using
minimal membrane
model tocould
explore
theasymmetrically
role of electrostatic
wesituation
are aware
that negatively
this is a major
charged lipids in the inner side kept at neutral pH and neutral lipids in the outer monolayer facing
simplification. In fact, the outer membrane of E. coli is a heterogeneous mixture of lipopolysaccharides
an acidic solution (representing the titration of the lipid polar heads at low pH [27]). Although this is
(LPS) anda convenient
phospholipids
[28] that differs significantly from plasma membranes [29]. The outer leaflet
minimal model to explore the role of electrostatic interactions, we are aware that this is
is almost aexclusively
comprisedInoffact,
LPS,the
which
a very effective
barrier
for the spontaneous
major simplification.
outerare
membrane
of E. coli
is a heterogeneous
mixture diffusion
of
of lipophilic
compounds due
to and
the phospholipids
low fluidity of
the
LPS
hydrocarbon
domain
and the
strong lateral
lipopolysaccharides
(LPS)
[28]
that
differs
significantly
from plasma
membranes
[29].between
The outerLPS
leaflet
is almost[28].
exclusively
comprised
which are
a very effective
for
interactions
molecules
The inner
leafletofisLPS,
a mixture
of cardiolipin
andbarrier
phospholipids
the
spontaneous
diffusion
of
lipophilic
compounds
due
to
the
low
fluidity
of
the
LPS
hydrocarbon
(PG and PE) so that charged lipids are in a similar proportion to that found in endoplasmatic reticulum
domain and the strong lateral interactions between LPS molecules [28]. The inner leaflet is a mixture
membranes [28,29].
of cardiolipin and phospholipids (PG and PE) so that charged lipids are in a similar proportion to
Figure
shows
that asymmetric
conduction
can [28,29].
be achieved in symmetrical neutral membranes
that1found
in endoplasmatic
reticulum
membranes
(PC||PC, Figure
and asymmetrically
chargedcan
membranes
Figure
1b)membranes
when there is a
Figure1a)
1 shows
that asymmetric conduction
be achieved(PC||PS,
in symmetrical
neutral
(PC||PC,
Figure
1a) and
asymmetrically
charged
membranes
Figure
1b) when
thereagreement
is a
realistic pH
gradient
across
the
channel. The
PNP-3D
model (PC||PS,
calculations
show
a good
realistic
pH
gradient
across
the
channel.
The
PNP-3D
model
calculations
show
a
good
agreement
not only with the experimental recordings shown in the inset of each figure but also with not
previous
only with the experimental recordings shown in the inset of each figure but also with previous
experimental
studies [12,30]. In neutral membranes (Figure 1a), both the total current and the current
experimental studies [12,30]. In neutral
membranes (Figure 1a), both the total current and the
carried by
the prevailing counterions (Cl− ions in this case) are asymmetric. Figure 1b shows the
current carried by the prevailing counterions (Cl− ions in this case) are asymmetric. Figure 1b shows
importance
membrane
charges. In
asymmetrical
membranes,
the current
rectification
is is
slightly
the of
importance
of membrane
charges.
In asymmetrical
membranes,
the current
rectification
higher, but
the major
thatchange
now the
current
is dominated
by cations,
in contrast
to thetoresults
slightly
higher,change
but the is
major
is that
now the
current is dominated
by cations,
in contrast
results1a.
shown
in change
Figure 1a.inThis
in the
channel selectivity
can be crucial
the resistance
acid
shown intheFigure
This
thechange
channel
selectivity
can be crucial
for theforacid
resistance
mechanisms
of
the
bacteria,
requiring
a
precise
control
of
positively
charged
protons
[10].
mechanisms of the bacteria, requiring a precise control of positively charged protons [10].
Figure 1. (a) Calculated current-voltage (I-V) curve of OmpF channel in the 3||7 configuration for
Figure 1. (a) Calculated current-voltage (I-V) curve of OmpF channel in the 3||7 configuration for
symmetrical neutral membranes PC||PC; (b) Calculated current-voltage (I-V) curve of OmpF
symmetrical neutral membranes PC||PC; (b) Calculated current-voltage (I-V) curve of OmpF channel
channel in the pH 3||7 configuration for asymmetrical membranes PC||PS. KCl concentration is
in the pH25||25
3||7 mM.
configuration
for asymmetrical
membranes
concentration
is 25||25
The insets show
a representative
experimentalPC||PS.
I-V curveKCl
of OmpF
single channel
at the mM.
The insetssame
show
a
representative
experimental
I-V
curve
of
OmpF
single
channel
at
the
same
conditions
conditions as that of the main figure.
as that of the main figure.
Entropy 2017, 19, 116
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The calculations provide new insights about the structural origin of current rectification. Figure
2a compares
the electrostatic
across
OmpForigin
channel
for uncharged
The calculations
provide newpotential
insights about
the the
structural
of current
rectification.(left)
Figureand
2a
asymmetrically
charged
membrane
(right).
When
the
membrane
is
neutral,
the
titration
of acidic
compares the electrostatic potential across the OmpF channel for uncharged (left) and asymmetrically
residues membrane
in the channel
the acidic
solutionthe
creates
an of
inhomogeneous
charged
(right).mouth
Whenfacing
the membrane
is neutral,
titration
acidic residues charge
in the
distribution
yielding
an
overall
positive
potential,
that
is,
the
channel
is
anion-selective
shownan
in
channel mouth facing the acidic solution creates an inhomogeneous charge distributionas
yielding
Figure
1a.
Contrariwise,
when
negative
charges
are
placed
in
the
membrane
facing
the
neutral
overall positive potential, that is, the channel is anion-selective as shown in Figure 1a. Contrariwise,
solution,
the balance
andinnow
the electrostatic
in solution,
the system
more negative,
when
negative
chargeschanges
are placed
the membrane
facingpotential
the neutral
theisbalance
changes
meaning
that
the
channel
prefers
positive
ions
as
depicted
in
Figure
1b.
Of
note,
the membrane
and now the electrostatic potential in the system is more negative, meaning that the channel
prefers
charge
not
only
regulates
the
electrostatic
potential
in
the
corresponding
channel
mouth,
but it
positive ions as depicted in Figure 1b. Of note, the membrane charge not only regulates the electrostatic
affects the
system. From
the PNP
system
equations
we also
obtained
potential
in whole
the corresponding
channel
mouth,
but itofaffects
the whole
system.
From three-dimensional
the PNP system of
diagrams
displaying
the
trajectories
of
ions
through
the
channel.
Figure
2b
displays
an through
examplethe
of
equations we also obtained three-dimensional diagrams displaying the trajectories
of ions
this complete
view foran
a example
monomer,
at the
same conditions
in Figure at
1bthe
(KCl
25||25
mM,
channel.
Figure3D
2b displays
of this
complete
3D view foras
a monomer,
same
conditions
neutral||negative
membrane,
pH
3||7)
and
no
applied
voltage.
Remarkably,
protons
flow
down
the
as in Figure 1b (KCl 25||25 mM, neutral||negative membrane, pH 3||7) and no applied voltage.
− and H+ from left to right and K+ from right to left) in
pH
gradient,
inducing
the
flow
of
salt
ions
(Cl
Remarkably, protons flow down the pH gradient, inducing the flow of salt ions (Cl− and H+ from left
order
toand
fulfill
requirements.
The importance
of membrane
charges
is importance
graphically
to
right
K+electroneutrality
from right to left)
in order to fulfill
electroneutrality
requirements.
The
demonstrated
by
the
way
in
which
the
cation
streamlines
bend
at
the
pore
exit
trying
to
approach
to
of membrane charges is graphically demonstrated by the way in which the cation streamlines bend
+ and Cl− ions follow
the
negatively
charged
lipid
heads.
Figure
2b
demonstrates
as
well
that
K
at the pore exit trying to approach to the negatively charged lipid heads. Figure 2b demonstrates as
well-separated
trajectories
when crossing
the channel
[31],when
a characteristic
mechanism
of
well
that K+ andscrew-like
Cl− ions follow
well-separated
screw-like
trajectories
crossing the
channel [31],
some
bacterial
porins
that
is
not
lost
despite
the
large
pH
gradients
used
in
our
study
[11].
This
a characteristic mechanism of some bacterial porins that is not lost despite the large pH gradients
peculiarity
also [11].
suggested
in the electrostatic
maps around
central constriction
of the
used
in our is
study
This peculiarity
is also suggested
in thethe
electrostatic
maps around
thechannel
central
in
Figure
2a.
constriction of the channel in Figure 2a.
Figure
units across
across OmpF
OmpF for
for an
an uncharged
uncharged membrane
Figure 2.
2. (a)
(a) Electrostatic
Electrostatic potential
potential in
in kT/e
kT/e units
membrane (PC||PC,
(PC||PC,
left)
and
an
asymmetrically
charged
membrane
(PC||PS,
right);
(b)
3D
ion
current
paths
along
left) and an asymmetrically charged membrane (PC||PS, right); (b) 3D ion current paths along the
the
OmpF
Ions are
are represented
OmpF channel
channel (in
(in grey)
grey) for
for an
an asymmetrically
asymmetrically charged
charged membrane
membrane (PC||PS).
(PC||PS). Ions
represented as
as
streamlines
followed
on on
average
by each
ion, not
into account
the fluxthe
values.
streamlines to
toshow
showthe
thepaths
paths
followed
average
by each
ion,taking
not taking
into account
flux
Figures
obtained
PNP-3D
and represented
Mayavi,using
a Python-based
values. were
Figures
were from
obtained
fromcalculations
PNP-3D calculations
and using
represented
Mayavi, a
program for 3D representation. The conditions used were KCl 25||25 mM, pH 3||7, and no applied
Python-based program for 3D representation. The conditions used were KCl 25||25 mM, pH 3||7,
electrostatic potential through the system.
and no applied electrostatic potential through the system.
3.2.
Structural Basis
3.2. Structural
Basis of
of Ion
Ion Pumping
Pumping
The
feature of
of large
largeβ-barrel
β-barrelchannels
channelssuch
suchasasOmpF
OmpFwith
with
respect
Na-channels,
The distinctive
distinctive feature
respect
to to
Na-channels,
K
K
channels
channelsis isthat
thatthey
theydo
donot
not transport
transport aa unique
unique chemical
chemical species,
species, but
but allow
channels
or or
ClClchannels
allow the
the
simultaneous
transport of
of all
all of
of them
simultaneous transport
them present
present in
in the
the solution
solution [32].
[32]. Hence,
Hence, when
when aa salt
salt concentration
concentration
gradient
is
imposed,
both
anions
and
cations
flow
from
the
concentrated
to
the
diluted
gradient is imposed, both anions and cations flow from the concentrated to the diluted side
side of
of the
the
membrane
but
contribute
differently
to
the
overall
current
as
imposed
by
the
channel
selectivity.
membrane but contribute differently to the overall current as imposed by the channel selectivity.
Figure
3a shows
shows structure-based
structure-basedcalculations
calculationsofofthe
theI-V
I-Vcurve
curve
considering
a salt
gradient
> ctrans
Figure 3a
considering
a salt
gradient
ccis c>cisctrans
and
and
the asymmetrically
charged
membrane.
We can
observe
channel
is selective
to cations,
the asymmetrically
charged
membrane.
We can
observe
thatthat
the the
channel
is selective
to cations,
in
in
excellent
agreement
with
the
electrophysiological
recordings,
which
are
shown
in
the
inset
of
the
excellent agreement with the electrophysiological recordings, which are shown in the inset of the
figure
Notably, the
restrains the
the ability
ability of
of the
figure [7].
[7]. Notably,
the existence
existence of
of aa concentration
concentration gradient
gradient restrains
the channel
channel to
to
rectify
but the
the I-V
I-V curve
curve still
still displays
displays aa significant
significantasymmetry
asymmetryfor
forthe
thetotal
totaland
andKK++current.
current.
rectify the
the current,
current, but
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+
Figure
3b shows
the the
concentration
Cl−−ions
ionsin in
concentration
Figure
3b shows
concentrationprofile
profileof
of KK+ and
and Cl
thethe
pHpH
andand
concentration
conditions
of
Figure
3a,
but
at
two
different
polarities:
+200
and
−
200
mV.
Positive
voltages
promote
conditions of Figure 3a, but at two different polarities: +200 and −200 mV. Positive voltages promote
+
−
+
−
the downhill
transport
of Kwhile
while
hinderingthe
the transport
transport of
that
could
hardly
reach
the the
the downhill
transport
of K
hindering
ofCl
Cl ions
ions
that
could
hardly
reach
−, but
− , but
diluted
In contrast,
negative
voltagespermit
permitthe
the downhill
downhill transport
at the
time time
diluted
side.side.
In contrast,
negative
voltages
transportofofClCl
at same
the same
+ that does not differ much from that found at positive voltages.
provide
a dominant
current
of+Kthat
provide
a dominant
current
of K
does not differ much from that found at positive voltages.
Figure
3. (a)3.Calculated
current-voltage
channelininthe
the
3||7
configuration
Figure
(a) Calculated
current-voltage(I-V)
(I-V)curve
curve of
of OmpF
OmpF channel
3||7
configuration
for for
concentration
ratio
r
=
10
(KCl
250||25
mM)
and
asymmetrical
membranes
PC||PS.
The
inset
shows
concentration ratio r = 10 (KCl 250||25 mM) and asymmetrical membranes PC||PS. The inset shows
an experimental
curve
at the
sameconditions
conditions as
as in the
(b)(b)
Concentration
profiles
for for
an experimental
I-V I-V
curve
at the
same
themain
mainfigure;
figure;
Concentration
profiles
potassium
and chloride
across
OmpF
channel
gray)
underan
anapplied
appliedpotential
potential of +200
+200 mV
potassium
and chloride
ionsions
across
the the
OmpF
channel
(in(in
gray)
under
mV, as indicated.
The conditions
for the
calculations
thesame
sameas
asin
in (a).
(a).
and −mV
200and
mV,−200
as indicated.
The conditions
usedused
for the
calculations
areare
the
To evaluate if the asymmetry observed in Figure 3 may lead to an actual pumping of
To
evaluate if
the asymmetry
observed
Figure 3types
mayoflead
to an actual
pumping
of counterions
counterions
(cations
in this case),
we usedindifferent
zero-mean
fluctuation
potentials.
The
(cations
in this case),
wesignals
used different
types
of zero-mean
oscillating
voltage
oscillating
voltage
that appear
in living
cells mayfluctuation
come frompotentials.
dissipative The
processes
yielding
signals
that appear
living
dissipative
thermal
noise
thermal
noise [2]inbut
couldcells
alsomay
havecome
other from
origins,
such as theprocesses
stochastic yielding
opening and
closing
of [2]
membrane
channels
in
the
membrane
or
the
occurrence
of
non-equilibrium
chemical
reactions.
but could also have other origins, such as the stochastic opening and closing of membrane channels
these
signals are simulated
using random
telegraph
noise (RTN)
with different
in theTypically,
membrane
or later
the occurrence
of non-equilibrium
chemical
reactions.
Typically,
these later
voltage
distributions
[33,34].
In
particular,
we
used
voltages
fluctuating
between
two
extreme
values
signals are simulated using random telegraph noise (RTN) with different voltage distributions
[33,34].
and also following a Gaussian and a random distribution. Figure 4 (left panel) shows these input
In particular, we used voltages fluctuating between two extreme values and also following a Gaussian
signals and the outputs obtained, based on the I-V curve of Figure 3a.
and a random distribution. Figure 4 (left panel) shows these input signals and the outputs obtained,
Numerically solving the PNP system of equations allows us to obtain separately the current
basedoutput
on theofI-V
curve
3a.the left panels of Figure 4. In all cases, we consider that the output
each
ion, of
as Figure
shown in
Numerically
solving
the
PNP
system
of equations
allows us experimental
to obtain separately
signal is the slave
of the
input
one,
as shown
in the corresponding
situationsthe
[7]. current
As
output
of
each
ion,
as
shown
in
the
left
panels
of
Figure
4.
In
all
cases,
we
consider
that
the
output
could be expected from the cationic selectivity displayed in Figure 3, the total current and the
signalcurrent
is the slave
ofby
thecations
input are
one,pretty
as shown
in although
the corresponding
experimental
situations
[7].
As could
carried
similar,
Itotal is always
larger at negative
values
thanks
to the contribution
anions. selectivity
This is clearly
seen in theinright
panel3,ofthe
Figure
4, where
theand
voltage
be expected
from the of
cationic
displayed
Figure
total
current
the and
current
current
distributions
are displayed.
The distributions
currentlarger
are biased,
reflecting
the asymmetry
carried
by cations
are pretty
similar, although
Itotal is of
always
at negative
values
thanks to the
of the output
signals,
which
may imply
cations.
contribution
of anions.
This
is clearly
seen an
in eventual
the rightpumping
panel of of
Figure
4, where the voltage and current
The qualitative message of Figure 4 is appealing but a quantitative analysis is desirable. A more
distributions are displayed. The distributions of current are biased, reflecting the asymmetry of the
convenient way of probing the pumping of ions is calculating the average currents for both the total
output signals, which may imply an eventual pumping of cations.
current and the current carried by each ion, as shown in Figure 5 for each signal.
The qualitative
of Figure
4 is positive
appealing
butpotentials
a quantitative
analysis
is desirable.
The averagemessage
total current
turns from
at low
to negative
at high
potentials, A
formore
convenient
way
of
probing
the
pumping
of
ions
is
calculating
the
average
currents
for
both
the
all signals displayed. As mentioned before, this means that the superimposed fluctuating potential is total
current
and
the current
carried
each originated
ion, as shown
inconcentration
Figure 5 for gradient.
each signal.
high
enough
to outweigh
theby
current
by the
However, due to the
multiionic
character
of the channel,
one may
wonder
whether
the change
in the average
current
is
The
average
total current
turns from
positive
at low
potentials
to negative
at high
potentials,
assure the As
electrical
pumping
of ions
In the
case
Cl− ions, their average
current
is
for allenough
signalstodisplayed.
mentioned
before,
this[12].
means
that
theofsuperimposed
fluctuating
potential
always
negative,
which
indicates
that
these
ions
always
flow
downhill
(i.e.,
from
the
concentrated
to
is high enough to outweigh the current originated by the concentration gradient. However, due to
the dilutedcharacter
solution)of
regardless
the magnitude
of the applied
voltage.
However,
the behavior
of K+
the multiionic
the channel,
one may wonder
whether
the change
in the
average current
is
ions is completely different: they display a positive average current (<IK> > 0) at low potentials that
enough to assure the electrical pumping of ions [12]. In the case of Cl− ions, their average current is
turns into a negative current when the applied voltage is high enough. That is, they present an uphill
always negative, which indicates that these ions always flow downhill (i.e., from the concentrated to the
transport at sufficiently high potentials. Interestingly, the voltage at which <IK> < 0 depends on the
diluted solution) regardless the magnitude of the applied voltage. However, the behavior of K+ ions is
completely different: they display a positive average current (<IK > > 0) at low potentials that turns into
a negative current when the applied voltage is high enough. That is, they present an uphill transport
Entropy 2017, 19, 116
6 of 9
(a)
200
Cl
+
H & OH
100
1.0
300
Total
+
K
200
0.5
100
0
0
-100
-100
-200
-200
-300
-300
0.0
I (pA)
V (mV)
-
Total
+
K
-200 -100
0
100
I (pA)
1.0
0.5
Counts
300
Counts
at sufficiently high potentials. Interestingly, the voltage at which <IK > < 0 depends on the type of
signal,
indicating
that some signals are more efficient than others. Among those analyzed here,
Entropy
2017, 19, 116
6 of 9 the
most efficient is that operating at constant V. This could be the signal found when voltage oscillations
signal,
indicating
some
signals
are more efficient
than others.
Among
those analyzed
cometype
fromofthe
flickering
of a that
unique
type
of membrane
pore operating
between
well-defined
current
the most
is thatthe
operating
constant would
V. This occur
could be
the signal found
when
voltage
stateshere,
[6]. For
theseefficient
conditions,
actual at
pumping
at moderate
voltage
values
around
oscillations
come
from the
flickering ofmay
a unique
of membrane
operating
100−150
mV. Such
a voltage
configuration
not betype
the most
probable pore
one, having
in between
mind that in
well-defined current states [6]. For these conditions, the actual pumping would occur at moderate
the cell membrane there is actually a wide variety of diverse membrane channels that furthermore
voltage values around 100−150 mV. Such a voltage configuration may not be the most probable one,
usually display several subconductance states [9]. Voltage configurations depicting more realistic
having in mind that in the cell membrane there is actually a wide variety of diverse membrane
fluctuations
or randomly
ones)
require considerably
higher configurations
voltages to obtain
channels(Gaussian
that furthermore
usually distributed
display several
subconductance
states [9]. Voltage
<IK > <
0.
A
potential
larger
than
250
mV
seems
unlikely
to
appear
in
the
bacterial
membrane
(voltage
depicting more realistic fluctuations (Gaussian or randomly distributed ones) require considerably
fluctuations
are typically
between
100
mV and
+100
mV
but we
do nottoaim
to establish
a
higher voltages
to obtain
<IK> < 0.−A
potential
larger
than
250[10]),
mV seems
unlikely
appear
in the
bacterialconnection
membrane between
(voltage fluctuations
areintypically
between
−100 mV and
+100 mV
[10]),
quantitative
the situation
vivo and
the mechanism
depicted
here.
Webut
areweaware
dovitro
not aim
to establishand
a quantitative
connection shown
betweenhere
the situation
vivo
and the mechanism
that in
experiments
in silico calculations
provide in
only
a qualitative
picture that,
depicted
here.
We
are
aware
that
in
vitro
experiments
and
in
silico
calculations
shown
here
provide
although appealing, could differ from the actual one. On the other side, the conclusions of Figure 5
only
a qualitative
picture
that, although
appealing,tocould
from
the actualnoise.
one. On
thethe
other
are not
limited
to RTN,
but they
can be extended
whitediffer
noise
or thermal
For
sake of
side, the conclusions of Figure 5 are not limited to RTN, but they can be extended to white noise or
simplicity, here we use a triangular-shaped signal (see Figure 6) which gives essentially the same
thermal noise. For the sake of simplicity, here we use a triangular-shaped signal (see Figure 6) which
average as any random signal. Again, the distributed voltage means a lower efficiency of the pumping
gives essentially the same average as any random signal. Again, the distributed voltage means a
mechanism
so that of
voltages
around
300 mV so
are
required
obtain
transport
of K+toagainst
lower efficiency
the pumping
mechanism
that
voltagesto
around
300a mV
are required
obtain a their
concentration
gradient.
transport of
K+ against their concentration gradient.
0.0
1
2
3
4
1
2
t (s)
3
4
-300
0
300
V (mV)
t (s)
600
σ = 250 mV
Cl+
H & OH
Total
+
K
400
0.5
200
0
0
0.0
I (pA)
V (mV)
200
1.0
Total
+
K
400
-450 -300 -150
0
150
I (pA)
-200
-200
1.0
-400
-400
0.5
-600
-600
1
2
3
4
1
2
t (s)
3
Counts
600
Counts
(b)
0.0
4
-800 -400
0
400
800
V (mV)
t (s)
Vmax = 300 mV
Cl
+
H & OH
300
1.0
Total
+
K
200
0.5
100
100
0
0
-100
-100
-200
-200
-300
-300
0.0
I (pA)
V (mV)
200
-
Total
+
K
-200 -100
0
100
I (pA)
1.0
0.5
Counts
300
Counts
(c)
0.0
1
2
3
t (s)
4
1
2
t (s)
3
4
-300 -150
0
150
300
V (mV)
Figure
4. Left
panel:
Representative calculated
calculated traces
input
signals
(constant
Figure
4. Left
panel:
Representative
traces showing
showingdifferent
different
input
signals
(constant
voltage
(a);
Gaussian-distributed
voltage
(b);
and
random
noise
(c)
and
the
corresponding
output
voltage (a); Gaussian-distributed voltage (b); and random noise (c) and the corresponding
output
currents; Right panel: Voltage and current (total and K+ current) distributions obtained from the
currents; Right panel: Voltage and current (total and K+ current) distributions obtained from the signals
signals as those shown in the left panel.
as those shown in the left panel.
Entropy 2017, 19, 116
7 of 9
Entropy 2017, 19, 116
Entropy 2017, 19, 116
7 of 9
7 of 9
t exp, V ctant
t exp, V ctant
Average
Averagecurrent
current(pA)
(pA)
20
20
0
0
Total
Total
K+
K+Cl
Cl+H & OHH+ & OH-
-40
-40
0
0
0
0
50
50
100 150 200 250
100 150 200 250
Total
Total
K+
K+Cl
Cl+H & OHH+ & OH-
-20
-20
-40
-40
t exp, V random
t exp, V random
20
20
0
0
-20
-20
-60
-60
t exp, V gauss
t exp, V gauss
20
20
0
0
50
50
Total
Total
K+
K+Cl
Cl+H & OHH+ & OH-
-20
-20
-40
-40
0
0
100 150 200 250
100 150 200 250
100
100
200
200
300
300
400
400
V (mV)
V (mV)
Figure
5. Average
Average
current
as
functionof
ofvoltage,
voltage, calculated
calculated
from
the
signals
showed
in Figure
Figure
4. 4.
Figure
5. Average
current
asas
a function
calculatedfrom
from
the
signals
showed
in Figure
Figure
5.
current
aa function
of
voltage,
the
signals
showed
in
4.
(b)
(b)
(a)
(a)
300
300
-
OHH & OH
100
100
100
100
0
0
0
0
-100
-100
-100
-100
-200
-200
-200
-200
-300
-300
-300
-300
0.5
0.5
1.0
1.0
t (s)
t (s)
1.5
1.5
0.5
0.5
1.0
1.0
t (s)
t (s)
20
20
200
200
1.5
1.5
0
0
I (pA)
I (pA)
VV(mV)
(mV)
200
200
300
300
Total
Total
K+
K+Cl
Cl+H & OHH+ & OH0
0
100
100
200
200
300
300
V (mV)
V (mV)
-20
-20
Average
Averagecurrent
current(pA)
(pA)
ClCl+
H+ &
Total
Total
+
K+
K
-40
-40
400
400
Figure 6. (a) Calculated traces showing a triangular input voltage and the corresponding output
Figure
6. Calculated
(a) Calculated
tracesshowing
showing aa triangular
triangular input
voltage
and
thethe
corresponding
output
Figure
6. (a)
traces
input
voltage
and
corresponding
output
current. The
The V
V and
and II distributions
distributions are
are almost
almost identical
identical to
to that
that obtained
obtained for
for aa random
random distribution
distribution
current.
current.
The
V
and
I
distributions
are
almost
identical
to
that
obtained
for
a
random
distribution
(right panel
panel of
of Figure
Figure 4b);
4b); (b)
(b) Calculated
Calculated average
average current
current as
as aa function
function of
of voltage,
voltage, from
from the
the signal
signal in
in
(right
(rightthe
panel
of Figure 4b); (b) Calculated average current as a function of voltage, from the signal in the
left
panel.
the left panel.
left panel.
In summary,
summary, our
our calculations
calculations show
show that
that the
the voltage
voltage requested
requested to
to obtain
obtain the
the uphill
uphill transport
transport of
of
In
counterions
is
significantly
increased
by
the
poor
selectivity
of
the
pore
allowing
the
downhill
counterions
is our
significantly
increased
the the
poorvoltage
selectivity
of the pore
allowing
downhill
In
summary,
calculations
showbythat
requested
to obtain
thethe
uphill
transport
transport of
of coions.
coions. Furthermore,
Furthermore, the
the oscillating
oscillating signals
signals characteristic
characteristic of
of membrane cells
cells (RTN and
and
transport
of counterions
is significantly
increased
by the poor selectivity
of themembrane
pore allowing(RTN
the downhill
thermal
noise)
display
relatively
low
efficiencies
so
that
remarkably
high
voltages,
at
the
limit
of
thermal
noise) display
relativelythe
lowoscillating
efficienciessignals
so that characteristic
remarkably high
voltages,
at the
limit
of
transport
of
coions.
Furthermore,
of
membrane
cells
(RTN
and
what
could
be
realistic,
would
be
necessary
to
observe
actual
electrical
pumping
of
ions
from
the
what could be realistic, would be necessary to observe actual electrical pumping of ions from the
thermal
noise)
display
relatively
low
efficiencies
so
that
remarkably
high
voltages,
at
the
limit
of
what
diluted side
side to
to aa more
more concentrated
concentrated one.
one. Our
Our analysis
analysis is
is limited
limited to
to certain
certain types
types of
of signals
signals but
but there
there
diluted
couldcould
be realistic,
would
becontributing
necessary to
observe
actual
electrical
pumping
ofpumping
ions from
the diluted
be
other
processes
to
the
channel
dynamics
that
facilitate
the
process.
It
could be other processes contributing to the channel dynamics that facilitate the pumping process. It
side to
a
more
concentrated
one.
Our
analysis
is
limited
to
certain
types
of
signals
but
there
could
be
has
been
shown
that
the
permeation
of
ions
could
be
enhanced
by
transient
fluctuations
in
the
has been shown that the permeation of ions could be enhanced by transient fluctuations in the
channel
shape.
Such
so-called
“breathing”
motions
may
play
a
key
role
in
channel
gating
as
voltage
otherchannel
processes
contributing
to
the
channel
dynamics
that
facilitate
the
pumping
process.
It
has
been
shape. Such so-called “breathing” motions may play a key role in channel gating as voltage
sensors
[35–37].
shown
that the
permeation of ions could be enhanced by transient fluctuations in the channel shape.
sensors
[35–37].
Such so-called “breathing” motions may play a key role in channel gating as voltage sensors [35–37].
Acknowledgments: Authors
Authors acknowledge
acknowledge the
the financial
financial support
support by
by the
the Ministry
Ministry of
of Economy
Economy and
and
Acknowledgments:
Competitiveness
of
Spain
(Project
No.
FIS2013-40473-P
and
FIS2016-75257-P),
and
Universitat
Jaume
I
(Project
Competitiveness Authors
of Spain acknowledge
(Project No. FIS2013-40473-P
and FIS2016-75257-P),
and
Jaume
I (Project
the financial support
by the Ministry
ofUniversitat
Economy and
Competitiveness
Acknowledgments:
No. P1.1B2015-28).
P1.1B2015-28).
No.
of Spain
(Project
No. FIS2013-40473-P and FIS2016-75257-P), and Universitat Jaume I (Project No. P1.1B2015-28).
Author
Contributions:
A.A.
and
M.Q.-M.conceived
conceived and
and designed
designed
the
calculations;
M.A.-A.
performed
the the
Author
Contributions:
A.A.
and
M.Q.-M.
designedthe
thecalculations;
calculations;
M.A.-A.
performed
Author
Contributions:
A.A.
and
M.Q.-M.
conceived
and
M.A.-A.
performed
the
numerical
calculations;
M.-L.L.
performed
the
experiments;
A.A.
and
M.Q.-M.
analyzed
the
data
and
wrote
the
numerical
calculations;
M.-L.L.
performed
the
experiments;
A.A.
and
M.Q.-M.
analyzed
the
data
and
numerical calculations; M.-L.L. performed the experiments; A.A. and M.Q.-M. analyzed the data and wrote thewrote
manuscript.
the manuscript.
manuscript.
Conflicts
of Interest:
Interest:
The
authors
declare
noconflict
conflictof
of interest.
interest.
Conflicts
of Interest:
TheThe
authors
declare
nono
Conflicts
of
authors
declare
conflict
of
interest.
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