Table Of ContentSelf-optimized biological channels in facilitating the transmembrane movement of
charged molecules
V.T.N. Huyen1 , Le Bin Ho2 , Vu Cong Lap1 , and V. Lien Nguyen1 ∗
1 Institute for Bio-Medical Physics, 109A Pasteur, 1st Distr., Hochiminh City, Vietnam
2 Hochiminh City Institute of Physics, VAST, Vietnam
Weconsiderananisotropicallytwo-dimensionaldiffusionofachargedmolecule(particle)through
a large biological channel underan external voltage. The channel is modeled as a cylinder of three
structureparameters: radius,length,andsurfacedensityofnegativechargeslocated atthechannel
interior-lining. Thesechargesinduceinsidethechannelapotentialthatplaysakeyroleincontrolling
6
theparticlecurrentthroughthechannel. Itwasshownthattofacilitatethetransmembraneparticle
1
movement the channel should be reasonably self-optimized so that its potential coincides with the
0
2 resonant one, resulting in a large particle current across thechannel. Observedfacilitation appears
to be an intrinsic property of biological channels, regardless the external voltage or the particle
n
concentration gradient. This facilitation is very selective in the sense that a channel of definite
a
structureparameterscanfacilitatethetransmembranemovementofonlyparticlesofpropervalence
J
atcorrespondingtemperatures. Calculationsalsoshowthatthemodeledchannelisnon-Ohmicwith
9
the ion conductance which exhibits a resonance at the same channel potential as that identified in
2
thecurrent.
] PACSnumbers: 87.15.hj,87.16.Dg,87.16.Vy,05.10.Gg
t
f
o
s I. INTRODUCTION very experimentally problemic due to the puzzled com-
.
t plexities related to both the channel structure and the
a
m measurement systems.
Biological channels are responsible for regulating the
- fluxesofionsandmolecules(hereafterreferredtoaspar- Theoretically, to describe the CFPM several models
d
ticles for short) across membranes and, therefore, are have been suggested. Considering the one-dimensional
n
o critically important for the cell functioning [1]. As well- (1D)diffusionmodelwithaposition-dependentdiffusion
c known, these protein channels are very efficient in the coefficient, Berezhkovskii et al. supposedly introduced a
[ sensethat they supportaveryfast,selective,androbust squarepotentialwell,spanningthewholechannellength,
across-membrane transport, regardless of environment that brings about a channel-particle interaction [10–13].
1
v fluctuations [2]. Surprisingly, such privileged properties It was then shown that at a given solute concentration
3 have been observed even in the case of large water-filled difference there exists an optimum potential well depth
6 channels, where the particle transport does not involve that can maximize the particle current, facilitating the
9 the use of metabolic energy or conformational changes channel function. In this model (i) the channel is as-
7
andwas assumedto be simply diffusive [3]. Understand- sumed large enough so that all the effects related to the
0
ing the nature of this channel-facilitated particle move- particle size can be omitted, (ii) a single-particle diffu-
.
1 ment(CFPM)iscruciallyimportantfromthefundamen- sion is considered, neglecting all many particle correla-
0 tal molecular biology as well as the application point of tions, and, particularly, (iii) no realistic potential was
6
view (Many modern drugs are developed in the way of assigned as the source for the square potential well in-
1
using the ion-channelsto enhance their efficiency, see for troduced. BauerandNadler considereda similar 1Ddif-
:
v example Refs.[4–6]). fusion model with a square potential well that is how-
i
X Experimentally, there are accumulative data showing everassociatedlocallywithonlythe particleboundtem-
r thatthe observedCFPMis reallyresultedfromsomein- porarily inside the channel [14]. Using the macroscopic
a teraction between the moving particle and the channel- versionofFick’sequation,itwasthendemonstratedthat
interior lining [7, 8]. Recent advancements of high- a transportincrease always occurs for any square poten-
resolution current recording enable single-channel mea- tialwells. However,as alreadynotedby the authors,the
surements that provide directly a living picture of how squarepotentialwellexploitedinthismodelisalsorather
anindividualchannelfunctionsand,therefore,shedlight crudeandamorerealisticpotentialshouldbefound[14].
on the characteristics of channel current in dependence From the very other point of view, Kolomeisky models
ondifferent(channelandenvironment)parameters[7–9]. the channel as a set of discrete binding sites arranged
However,revealingexactlythenatureofchannel-particle stochastically [15]. In such the discrete-state model the
interaction as well as the mechanism of CFPM is still particles are assumed to hop along the binding sites in
translocations across the channel and the optimum cur-
rent may be achieved depending on the spatial distri-
bution of binding-sites and the site-particle interactions
∗Correspondingauthor, onleavefromVAST/Institute of Physics, [15, 16]. This model is so simple that the main dynamic
Hanoi,email: [email protected] properties of the problem can be calculated exactly. It
2
was alsodemonstratedthat the discrete-statemodel [15] sponding temperatures. So, the model suggests that fa-
andthecontinuumdiffusionmodel[10]arecloselyrelated cilitatingthetransmembraneparticlemovementisanin-
andcanbe effectively mappedinto eachother[17]. Nev- trinsic property of biological channels. This property is
ertheless,like the squarepotential wellin the continuum independent of the external factors such as the exter-
models[10,14],the natureofthe bindingsites (akindof nal voltage or the bulk particle concentration gradient,
channel-particleinteraction)andthehoppingmechanism though these factors may strongly influence the magni-
of particles in the discrete-state model [15] still need to tude of various particle dynamical characteristics.
be identified. The paper is organized as follows. Sec.II introduces
Importantly, in all the models mentioned [10, 14, 15] the 2D diffusion model for the problem under study, in-
the channel-particle interaction (which was expressed cluding the motion equation with an exact expression
by a square potential well or a binding site) is gener- of the channel-induced potential, and describes the cal-
ally viewed as the crucial condition for the transmem- culating method. Sec.III presents the main numerical
brane transport to be facilitated (see also [18]). Note resultsobtained. These resultsarediscussedin greatde-
again that in these models the particle motion is merely tail, showing the influence of various factors on the par-
considered one-dimensional. Recently, Dettmer et al. ticle dynamical characteristics. A particular attention is
have measured the diffusivity of spherical particles in given to the self-optimized property of the channels in
closely-confining, finite length channels [19]. Measure- facilitating the transmembrane particle movement. The
ments demonstrated a strongly anisotropic diffusion in paper concludes with a brief summary in the last Sec.IV
the channel interior: while the diffusion coefficient par-
allel to the channel axis remained constant throughout
the entire channel interior, the perpendicular diffusion II. MODEL AND CALCULATING METHOD
coefficient showed an almost linear decrease from the
axis towards the channel wall. These observations put We consider a cylindrical channel of length L and
forward a need for the two-dimensional (2D) descrip- radius R that connects the two reservoirs with parti-
tion with direction-dependent diffusion coefficients when cle concentrations nL and nR as schematically drawn
studying the movement of particles inside a large chan- in Fig.1(a). The channel interior-lining carries negative
nel. Furthermore, experimentally, the single channel ki- charges which are for simplicity assumed to be continu-
netics was extensively studied at different external volt- ously and regularly distributed with a surface density σ.
ages [20, 21]. And, the experimental sublinear current- (The cation channels are believed to contain a net nega-
voltage (I-V) characteristics reported in Refs.[22, 23] is tive charge in the pore lining region of the protein [25].
oftenusedasoneofthebasicrequirementsfortheoretical In the case of potassium and gramicidin channels this
models [24]. is due to the partially chargedcarbonyloxygens [1, 25]).
In the present paper we consider a 2D diffusive move- Thesenegativesurfacechargescreateanelectrostaticpo-
ment of particles through a large water-filled channel, tential U which affects the movement of particles inside
takingintoaccountananisotropyofdiffusioncoefficients the channel. Particles are assumed to diffuse indepen-
as observed in Ref.[19] and an influence of external volt- dently, neglecting any many-particle correlation. In ad-
ageasdiscussedinRefs.[20,24]. The channelismodeled dition, the diffusivity of a particle inside the channel is
asacylindercharacterizedbythreestructureparameters: assumedanisotropicwiththetwodifferentdiffusioncoef-
radius, length, and surface density of negative charges ficients, Dz (parallelwith) andDx (perpendicular to the
of channel interior-lining. The potential created by this channel axis). Following Ref.[19], we assume that (i) Dz
charged interior-lining inside the channel is exactly cal- is constantthroughoutthe channelcylinder [0 x <R
≤| |
culated. Itcausesthe“channel-particleinteraction”that and 0 z L]and somewhatsmaller than the diffusion
≤ ≤
playsakeyroleinfacilitatingthetransmembraneparticle coefficient D0 in the bulk, Dz = αD0 with 0 < α < 1
movement. Solving the 2D stochastic Langevinequation [we choose in the present work α = 0.5 for definition]
for the model suggested we systematically analyze the and (ii) Dx linearly decreases as x going from the chan-
typically dynamical characteristics of particles such as nel axis (where the diffusion is isotropic) to the channel
the translocation probabilities, the translocation times, wall, Dx =[1 (x/R)]Dz.
− | |
the currents,andthe channelion conductance under the The model also involves a longitudinal voltage V, i.e.
influence of various factors: the channel-induced poten- the difference in electrical potential between the two
tial, the external voltage, or the difference in reservoir channel ends, that may include the intrinsic membrane
particle concentrations. It was particularly shown that potential[1]and/orsomeexternallyappliedvoltage[20].
to facilitate the transmembrane particle movement the This voltage drives the particles moving along the chan-
channel should be reasonably self-optimized with appro- nel. For definition, we assumed that the voltage V is
priate structure parameters so that its potential coin- directed from the left to the right [in Fig.1(a)] and the
cides withthe resonantone. Inaddition,this facilitation charge q carried by a particle is positive.
is very selective in the sense that a channel of definite Actually, due to the cylindricalsymmetry ofthe chan-
structure parameters can facilitate the transmembrane nel model suggested, the motion of a particle inside a
movement of only particles of proper valence at corre- channelcanbe effectivelydescribedbythe 2Dstochastic
3
differential equation(Langevin equationfor overdamped motion):
γzz 0 z˙(t) ∂zU(x,z) qV/L ξz(t)
= q + + , (1)
(cid:18) 0 γxx (cid:19)(cid:18)x˙(t)(cid:19) − (cid:18)∂xU(x,z)(cid:19) (cid:18) 0 (cid:19) (cid:18)ξx(t) (cid:19)
where R < x(t) < R and 0 z(t) L are the 2D-
− ≤ ≤
(a) (b) coordinatesoftheparticleatt-time,x˙ dx/dt,γxx(γzz)
≡
x → isthedragcoefficientinthex(z)-direction,U(x,z)isthe
V potential created by the charged channel lining, qV/L is
•q→ R UU((xx,,zz))((mmVV)) L = 5nm, R = 0.2nm the voltage-inducedforceactingonaparticleofchargeq
z
-50
nL L nR in the z-direction, and ξx(t) (ξz(t)) is the random force
-100 in the x (z)-direction which is as usual assumed to have
(c) x(nm)
a zero mean and a white noise correlation:
-0.2 0 0.2 -150
-50
U(0,z)(mV) -1-7050 0 1 2 3 4 5--215205 U(x,L/2)(mV) --225000 0 1 2 3 4 5 0 0.x1( 0n.m2) hItξνi(sth)ier=e w0orathnydthoξνm(et)nξtνio(tn′)tihe=St2oDkeνsδ-(Et−ints′t)e,inνre=laxti(,o2zn).
z(nm) between the diffusion coefficient D, the drag coefficient
z(nm)
γ, and the absolute temperature T of a medium: Dγ =
kBT, where kB is the Boltzmann constant.
FIG. 1: (color online) (a) Model of the cylindrical channel
In eq.(1) we need to identify the potential U(x,z) in-
under study; (b) Channel-induced potential U(x,z) of eq.(3)
side the channel. Within the model considered, as men-
is plotted for the channel with R =0.2 nm, L=5 nm, and
σ=−0.1C/m2 [Note: U(x,z)issymmetricalwithrespectto tioned above, U is the electrostatic potential created by
thesignofx];(c)theU(0,z)-potentialwell(red-solidline,see thechargedliningofacylindricalchannel. Bysolvingthe
theleftandbottomaxes)andtheU(x,L/2)-potentialbarrier fundamentalelectrostaticproblemforachargedcylinder
(blue-dashedline,seetherightandtopaxes)forthepotential offinite sizes,wecanexactlyderiveananalyticalexpres-
U(x,z) in (b). The potential U0 ≡U(0,L/2)≈−90.6 mV in sionof U as a function the (x,z)-coordinates[0 x<R
≤
this case. and 0<z <L]:
Rσ π z+ (x R)2+z2
U(x,z) = (1 + )ln[ − ]
πǫ0ǫ{ 2 z L+ p(x R)2+(z L)2
− − −
p
z+ (x+R)2+z2
ln[ ] , (3)
− z L+ p(x+R)2+(z L)2 }
− −
p
where R, L, and σ are the channel structure parameters ple, U(0,z) as a function of z (bottom and left axes)
definedabove,ǫ isthe vacuumpermittivity,andǫisthe and U(x,L/2) as a function of x (top and right axes) in
0
dielectricconstantofthewaterintheinteriorofthechan- Fig.1(c)]. While the well shape of the channel potential
nel [26]. Note that the potential U(x,z) is symmetrical U(x,z) in the z-direction directly affects the movement
with respect to the sign of x. of particles across the channel (as will be seen below),
its barrier shape in the x-direction demonstrates a no-
As anexample, Fig.1(b)showsthe potential U(x,z)of ticeable role of the transverse motion in the anisotropic
eq.(3) for the channel with R=0.2 nm, L=5 nm, and 2D-diffusion model considered.
σ = 0.1 C/m2. At a given x-coordinate, U(z) behaves
−
as a symmetrical potential well with the absolute mini-
mum at z =L/2. On the contrary,givena z-coordinate,
the U(x)-curve describesa symmetricalpotentialbarrier As a consequence of the observed symmetrical shape,
with the absolute maximum at x = 0 [see, for exam- the potential U(x,z)canbe characterizedby its valueat
4
the center of the channel, (x=0,z =L/2), where stepistakentobe∆t=0.0005τ ,whichisbelievedsmall
0
enough. Thedynamicalquantitiesweareinterestedin,as
Rσ √4R2+L2+L mentionedabove,includethetranslocationprobabilities,
U(0,L/2) = ln[ ] U . (4)
0
2ǫ0ǫ √4R2+L2 L ≡ the averagetranslocationtimes, the net particle current,
−
and the ion conductance.
This potential value U is uniquely determined by the
0 Actually, the calculating method we exploit in this
channel structure parameters (L, R, and σ) and can be
study is the Brownian dynamics. By solving the
used to characterize the potential U(x,z) on the whole:
Langevin equation, this method is rather appropriate
each channel creates a unique U(x,z) and each U(x,z)
for the problem of interest. A systematical classification
has a unique U . As an intrinsic characteristics of the
0 of computationalapproachesproposedand employedfor
channel, the quantity U will be used below as a typ-
0 studies of ion channels can be found in the review paper
ical measure of the channel potential U(x,z). Fig.1(c)
[30]. Here, in solving numerically eq.(1), for convenience
indicates the potential U 90.6 mV for the channel
potential U(x,z) examine0d≈in−this figure. we choose L as the unit of length, τ0 = L2/D0 as the
unit of time, and kBT as the unit of energy. So, for ex-
Thus, as an extension of the model suggested by ample, if L = 5 nm and D = 3.10−10m2/s [17], then
Bgueirsehzehdkobvysktihieetmaali.n[1f0a–c1t3o]r,stahsefpolrleoswens:t (mi)odtheleidsiffduisstiionn- τ0 ≈ 8◦.3 10−8s. Remind tha0t kBT ≈ 8.617 10−5 eV for
T =1 K.
is anisotropically two-dimensional (see eq.(1)), (ii) the
negatively charged channel interior-lining creates inside
the channel a potential that leads to the first term in
III. NUMERICAL RESULTS AND
the right hand of eq.(1) and that can be exactly identi-
DISCUSSIONS
fied as a function of only channel structure parameters
(see eq.(3)), and (iii) the external voltage causes a driv-
ing force expressedby the secondterm in the righthand Inpresenting simulationresults we introduce for short
of eq.(1). Further, the study will be focused on show- the symbols u0 qU0/kBT (referred to as the effective
≡
inghowthesefactorsaffectthedynamicalcharacteristics channel potential) and v qV/kBT (referred to as the
≡
of particles moving through the channel. The dynami- effectiveexternalvoltage). So,thedefinedparametersu0
cal characteristics we are here interested in include the and v also containthe particle chargeq and the medium
translocation probabilities, the translocation times, the temperature T. We should keep this in mind when dis-
particle current, and the ion conductance. To calculate cussing the role of the channel potential in facilitating
thesequantitieswehavetosolveeq.(1). Reasonably,this thetransmembraneparticlemovement. Additionally,for
stochastic equation can be solved numerically by using definition, in all the figures relating to the translocation
the molecular dynamics method [27]. probabilitiesandthe averagetranslocationtimes the pa-
A particle enters the channel from either the left (z = rameters R and nL(R) are kept constant: R = 0.04 (in
0) or the right (z =L) at random with the probabilities unit of L) and nL(R) = 145(15) mM [1]. Influences of
proportional to the reservoir particle concentration nL these parameters will be later discussed when analyzing
or nR, respectively [Fig.1(a)]. The initial x-coordinate the net current [Fig.6].
( R < x(t = 0) < R) and the initial velocity compo- Letus firstexamineobtainedresultsforthe transloca-
−
nents (z˙(0) and x˙(0)) are randomly given, following the tion probabilities which are separatelycalculated for the
standard molecular dynamics simulation procedure [27]. particlesmovingthroughthechannelfromthelefttothe
Started from the given initial conditions, a discrete tra- right(PL)andforthosemovingintheoppositedirection
jectory of the particle is step by step constructed. Given (PR) [see Fig.1 with the V-directionindicated]. In simu-
thechannelpotentialU(x,z)andtheexternalvoltageV, lations, the probability PL (or PR) is determined as the
ineachtime-step(∆t)therandomforces,ξx(z),areinde- ratio of the number of particles that passed through the
pendently generated and then the final coordinates and channelto the totalnumber ofparticlesthat enteredthe
velocity of the particle are determined from eq.(1) using channel from the left (or right).
the well-known Euler scheme [28, 29]. In the x-direction Fig.2 shows PL (blue dash-dotted lines) and PR (red
the full reflection condition is applied every time when dashedlines)plottedversusu forthe channelsatdiffer-
0
a particle runs into the channel wall, x = R. In the ent effective voltages v: 0( ), 2( and ), and 4( and
± × ◦ ⋄ •
other direction,oncethe z-coordinateis outofthe range ). Generally,thisfiguredemonstratesthatwithincreas-
∗
[0,L], the data for the simulated particle is fixed and ing u0 both the translocation probabilities, PL and PR,
this particle is no longer followed. The next particle en- increase steadily first [see main figure] and then become
ters the channelandundergoesadiffusion processinthe saturated [see the inset]. Such the PL(R)-versus-u0 be-
same way as described above. The number of particles haviorisobservedatanyvoltagev. Inthe caseofzerov,
involved in getting each of average values of studied dy- duetotheleft-rightsymmetryofthepotentialU(x,z)of
namical quantities is so large that for allthe data points eq.(3)thetwocurves,PL andPR,aretotallycoincidental
presented below the error bar nowhere exceeds the sym- and the common curve may be in a qualitative compari-
bolsize[ 105to107particlesdependingonthequantity son with Fig.3 in Ref.[10](where the considered diffusion
≈
and the direction of movement investigated]. The time isone-dimensionalandthepotentialwellissquare). Note
5
R) 1 ) 0 14 1
es PL( 0.8 0001...468 τmes ( 11 802 −−−⋅−−−⋅− τ τRL :: vv == 00((+x)),, 22((◊o)),, 44((∗•)) 0.8 −−−⋅−−−⋅− τ τRL :: uu00 == 11((+x)),, 22((◊o)),, 44((∗•))
babiliti 0.6 0 4 8 12 16 20 00.2 ans. ti 246 (a) 00..46 (b)
o Tr 0 0.2
pr 0 2 4 6 8 10 0 2 4 6 8 10
on 0.4 u0 v
ati
oc 0.2 FIG. 4: (color online) Translocation times τL (blue dash-
nsl dotted lines) and τR (red dashed lines) are plotted versus u0
a atdifferentv: 0(×and+), 2(◦and⋄),and4(•and∗)(a)and
Tr 0 τL(R) versus v at different u0: 1(× and +), 2(◦ and ⋄), and
1 2 3 4 5 6 7 8 9 10 4(• and ∗)] (b). Otherparameters are thesame as in Fig.2.
u
0
are of equal value [two corresponding curves are started
FIG. 2: (color online) Translocation probabilities PL (blue from the same point]. With increasing v the probabil-
dash-dottedlines) and PR (red dashed lines) are plotted ver- ity PL smoothly rises, while the probability PR strongly
sus u0 ≡ qU0/kBT for the channels of the same R, nL, descends. At v 5 the probabilities PR become practi-
and nR, but at different voltages v ≡ qV/kBT: 0(×), 2(◦ cally vanished fo≥r all the channels under study [no par-
and ⋄), and 4(• and ∗). The points are the simulation re-
ticle can move through the channel in the right-to-left
sults, whereas the lines are drawn as a guide for the eyes
direction]. The probability PL, on the contrary, contin-
[R=0.2 nm, L=5 nm, nL =145 mM,/nR =15 mM].
ues to grow with the tempo that gradually slows down
athigherv. Calculationsrevealthatevenatv =100the
0.8 channels are still not perfectly transparent for the posi-
R) tivelychargedparticlesmovingalongtheexternalvoltage
PL( 0.7 PL: u0 = 1(x), 2(o), 4(•) direction[PL =0.98or0.95foru0 =4or1,respectively].
es 0.6 Next, weconsideranotherfundamentalcharacteristics
abiliti 0.5 -prtohbeaabvileirtaiegsePtLra(Rns)lsotcuadtiieodnitnimFeig.sI.n2-a3c,cwoerdsaepnacerawteitlyhctahle-
b culated the average translocation times for the particles
o 0.4
pr moving through the channel from the left to the right
on 0.3 (τL)andfor thosemovinginthe opposite direction(τR).
ati In simulations, we count the time each of simulated par-
oc 0.2 ticles spends inside the channel. The average transloca-
nsl 0.1 P : u = 1(+), 2(◊), 4(∗) tion time τL (or τR) is then obtained by averagingthese
a R 0 spendingtimes overallthe particlesthatpassedthrough
r
T
0 the channel from the left to the right (or from the right
0 1 2 3 4 5 6 7 8 9 10
to the left).
v
Fig.4 shows how obtained translocation times τL(R)
vary with the effective potential u [Fig.4(a)] or the ef-
0
FIG. 3: (color online) Translocation probabilities PL (blue fective voltage v [Fig.4(b)]. Interestingly, in all the cases
dash-dottedlines) and PR (red dashed lines) are plotted ver- studiedinboththefigures,Fig.4(a)andFig.4(b),thetwo
sus v for the channels with different effective potentials u0: points, corresponding to τL and τR, are practically coin-
1(× and +), 2(◦ and ⋄), and 4(• and ∗). Other parameters cided. So, our2D-simulationssuggesta generalequality,
and symbols are thesame as in Fig.2. τL(u0,v)=τR(u0,v), that should be always valid in the
model studied regardlessof the shape of the channel po-
tential U(x,z) as well as the presence of the external
that with the chosen direction of V [Fig.1, q is positive] voltage V. This really causes some surprise, noting on
the external voltage raises PL (two higher curves) while the directed influence of the voltage V. Actually, a sim-
suppressingPR (twolowercurves),comparedtothe case ilar equality of the two average translocation times has
of v =0 (the middle curve). been previously suggested in Ref.[11], but it was there
The external voltage effects can more clearly be seen relating to the 1D diffusion model without any external
in Fig.3 where the probabilities PL(R) are presented as voltage. Fig.4 thus allows us to deal with the two times
the functions of v for the channels with different u0: τL andτR asasingleaveragetranslocationtimethatwill
1( and +); 2( and ); and 4( and ). At zero v be below denoted simply by τ.
× ◦ ⋄ • ∗
the two probabilities PL(R) associated to the same u0 The fact that the channel potential u0 raises the
6
4
12.10
(a)
4
4 10.10
11)) 10.10
--ττ00 8.104 8.104
nt I(nt I( 6.104 6.104
ee 4
CurrCurr 24..11004 4.104
0 1) 2.104
1 -0 u
τ m
v 2 3 8 10 nt I( 0 0 2 4 6 8 10
4 4 6 e
5 0 2 u0 rr
u 4
C 5.10
(b)
4
4.10
4
3.10
FIG.5: (color online) 3D-plot ofthecurrentI asafunction
4 n /n = 145/5
ofu andv. NoteontheresonantbehavioroftheI versusu 2.10 L R
0 0 145/10
curvesatdifferentvoltagesv. Otherparametersarethesame 4
1.10 145/15
as in Fig.2
0 5/140
u
-1.104 m
translocation time τ in Fig.4(a), while it also raises the 0 2 4 6 8 10
translocation probabilities in Fig.2, might cause some u
0
surprise. Actually,aswillbeseenbelow,itturnsoutthat
a competition between these two seemingly contrary ef-
fectsofthepotentialu leadstothemostimportantphe-
0
nomenon in the ion-channel physics - the CFPM. Com- FIG. 6: (color online) Resonant channel potential u0 = um
paring the points from three curves with different volt- is an intrinsic characteristics of the channel. (a) I versus u0
curves extracted from Fig.5 for some values of v [from bot-
ages v, we learn that in the region of large u [u 6
0 0 ≥ tom: v = 0, 1, 2, 3, 4, and 5]; All these curves show their
in Fig.4(a)] the time τ decreases almost linearly as v in-
creasesfrom0to4. Inawiderrangeofv,Fig.4(b)shows maximum at the same resonant potential, u0 = um (indi-
cated by the arrow). (b) I as a function of u at v = 2
0
thatthelargertheeffectivepotentialu ,thestrongerthe
0 for various values of the ratio nL/nR [from top: nL/nR =
relative effect ofv on τ becomes. In the limit ofhigh ex- 145/5, 145/10, 145/15, and 5/140]; The resonant potential
ternal voltagewhen the v-induced driving force becomes u0 =um (indicatedbythearrow)isindependentofreservoirs
to dominate the right hand in eq.(1), the translocation particleconcentrationratioandcoincideswithumdetermined
time should depend on v as τ 1/√v. in Fig.6(a). Note: in the case of nL/nR =5/140 the current
∝
Whilethequestionoftheparticulartimethatmostrel- isnegative(flowingfromrighttoleftinFig.1(a))andreaches
evantly describes the transmembranetransportandthat thelargest magnitude at thesame u0 =um.
can be directly measured is still under discussion [14],
the net currenthas alwaysservedas the most important
quantitythatshouldbe determinedtheoreticallyinclose plies that for givenq and T,to successfully facilitate the
comparison with experimental measurements. For the transmembraneparticlemovementthe channelhastobe
problem under study, the net particle current is deter- optimized with the appropriate structure parameters so
mined as the average number of particles the two reser- thatitspotentialU(x,z)coincideswiththeresonantone.
voirsactuallyexchangedviathechannelinaunitoftime For example, for q = 1 and T = 300K, to own the reso-
(τ0). nant potential of um = 3.5 as seen in Fig.5, the channel
Fig.5presentsa3D-plotofthecurrentI independence shouldbe self-optimized withthe followingstructure pa-
onthe effectivechannelpotentialu andtheeffectiveex- rameters: L = 5 nm, R = 0.2 nm, and σ = 0.1 C/m2
0 −
ternalvoltagev. Remarkably,contraryto themonotonic (given ǫ=80 [26]).
behaviors of PL(R) and τ in Figs.2-4,Fig.5 shows clearly To see whether the resonant potential um depends on
a resonant behavior of the current I: for a given voltage the externalvoltagev, we depict inFig.6(a) some I(u )-
0
v in the I versus u curve there always has an impres- curves extracted from Fig.5 at various v. Surprisingly,
0
sively absolute maximum at some resonant channel po- the resonant channel potential um (indicated by the ar-
tential, u0 = um. Remind that u0 qU0/kBT with U0 row)ispracticallythe sameforallthe curvesatdifferent
≡
uniquelydeterminedbythechannelstructureparameters voltages v. Actually, the fact that um is independent
(L, R, and σ). So, the maximum observed in Fig.5 im- of v can be seen right in Fig.5 for all the values of v
7
under study. Further, we check if the resonant poten- 4
12.10
tial um depends on another important external parame- (a)
ter, the difference in particle concentration between the 10.104
tfowrosreevseerravlovirasl.ueFsigo.f6(tbh)eprraetsieonntsL/thneR.I(uIn0)-tdheepceansdeesncoef -1)0 8.104 u 0 = 231.5
τ
1n2L./1niRn=Re1f4.[51/])5,,a1ll45th/1e0p,aarntdicl1e4c5o/n1c5e(nttarkaetnionfrogmradTiaenbtles nt I( 6.104 68
e
are directed along the external voltage V, i.e. from the urr 4.104
left to the right in Fig.1(a), and, therefore, the currents C
are always positive [see the higher three curves]. On the 2.104
contrary, in the case of the lowest curve in Fig.6(b) for
nL/nR = 5/140 (e.g. for K+-channels [1]), the particle 0
0 1 2 3 4 5
concentration gradient is directed from the right to the
v
left in Fig.1(a) and, consequently, the current becomes
4
negative (Note that in this case the concentrationgradi- 3.10
entisstrongwhiletheexternalvoltageisrelativelysmall, (b) 3.104 g
v =2). Importantly, all the curves for various nL/nR in
g
Fig.6(b) show the maximums in magnitude at the same ce 2.104 2.104 v
value of u that exactly coincides with the resonant po- n
0 a 12345
tential um determined in Fig.6(a). Thus, we arrive at ct
u
an important remark: at a given q and T, the resonant d v = 1
n
potentialis entirely determined by the channelstructure Co 1.104 2
3
parameters. It is an intrinsic property of the channels
5
and can not be affected by the external factors such as
the external voltage or the particle concentration gradi-
1 2 3 4 5 6 7 8 9 10
ent. u
0
Thus, in the present model, CFPM appears to be a
self-optimized property of biological channels: to facili-
tatethetransmembraneparticlemovement,thechannels FIG. 7: (color online) (a) The current I is plotted versus
shouldbe intrinsically optimizedwith appropriatestruc- the voltage v [I-V characteristics] for channels with different
potentials u (indicated in the figure). All I-V curves show
ture parameters. Additionally, facilitating the trans- 0
a sublinear behavior. (b) The channel ion conductance g as
membrane transport is very selective in the sense that
a function of the channel potential u at various voltages v
a channel of definite structure parameters can facilitate 0
(given in the figure). At any v the conductance always has
the transmembrane transport of only particles of proper the maximum at the same u0 = um ≈3.5. Inset: g versus v
valence at corresponding temperatures. for the channelwith resonant potential u0 =um.
Furthermore, we demonstrate in Fig.7(a) the current-
versus-voltage curves, I(v) (I-V) - characteristics, ex-
tracted from Fig.5 for several channels of different u . the current. So, Fig.7(b) gives one more demonstration
0
The highest curve describes the I-V characteristics of for the resonantly self-organized property of channels in
the resonantly self-optimized channel with u0 = um. It facilitating the transmembrane particle movement. The
is clear that all the I(v)-curves presented in this fig- v-dependenceofginthisfigureisrelatedtothesublinear
ure are nonlinear, indicating the non-Ohmic property behavior of the I-V curves in Fig.7(a) as just discussed
of the channel model studied. In this case, the channel above(seetheInsetinFig.7(b)). Notethatsuchthesub-
ion conductance, defined as the ratio of I to v [31], be- linearityofcalculatedI V curvesqualitativelyresembles
−
comesdependentontheappliedvoltage. TheI(v)-curves experimental data reported Refs.[22, 23].
in Fig.7(a) reveal that as v increases the conductances Finally, Fig.8 compares the I(u )-curves obtained for
0
g = I/v decrease fast first at small v, reach a minimum different Dx in showing the role of the transverse dif-
atv 2.5 3,andthenslightlyincreaseathigherv (see, fusion in the 2D-diffusion model under study. Four
≈ −
for example, the inset in Fig.7(b) for the case u0 =um). cases presented are: (1) Dx = 0, implying the 1D-
The voltage, where the conductance gets minimal, de- diffusion, (2) Dx = 0.5Dz, implying an anisotropically
pends on the potential u0 and the reservoirparticle con- 2D-diffusion with Dx constant and smaller than Dz, (3)
centrations. To look for a possible relation between the Dx =Dz,implyinganisotropically2D-diffusion,and(4)
channelionconductanceandtheresonantchannelpoten- Dx =[1 (x/R)]Dz used in this work (see correspond-
− | |
tial um associatedwith the current[Fig.6], wepresentin ing symbols given in the figure). Obviously, the curve
Fig.7(b) the conductances g calculated for channels of in the case of 1D-diffusion is largely separated from the
different potentials u at the same voltage v. Remark- rest,showinganessentialroleofthetransversediffusion.
0
ably, at any v the conductance g always has the maxi- In this limiting case there has also the maximum in the
mum at the same u0 =um as that identified in Fig.6 for I(u0)-curve, however, the current peak is lower and the
8
4 ductance. It was shown that while the external voltage
12.10
does not cause any especial effect, the channel poten-
4 tialincreasesboththetranslocationprobabilitiesandthe
10.10
average translocation times. And, surprisingly, studies
1) 8.104 demonstrated a single average translocationtime that is
-0 equally applied for the particles passing the channel in
τ
(
nt I 6.104 two contrary directions, regardless of even the directed
e influence of the external voltage.
urr 4.104 Dx = 0 Themostinterestingresultwasappearedinexamining
C = 0.5D
z the particle current. It was shown that at a given tem-
= D
2.104 = (1z-|x|/R)D perature the channel with appropriate structure param-
z eters can induce the resonant potential that effectively
0 facilitates the transmembrane movement of the particles
0 2 4 6 8 10 of a given valence, resulting in a very large net parti-
u cle current across the channel. In other words, to facil-
0
itate the transmembrane particle movement the channel
should be naturally self-optimized so that its potential
coincides with the resonant one. The resonant potential
FIG. 8: (color online) The I(u0)-curves for different Dx
isanintrinsiccharacteristicsofthechannelandfacilitat-
(indicated in the figure) are compared to show the role of
the transverse diffusion in the 2D-diffusion model considered ing the transmembrane particle movement is an intrin-
[v = 5, Dz = 0.5D0, other parameters are the same as in sic property of biological channels, independent of the
Fig.2]. external factors such as the external voltage or the par-
ticle concentration gradient. In addition, the observed
CFPMisveryselectiveinthesensethatachannelofdef-
resonant potential is much larger ( 8), compared to inite structure parameters can facilitate the transmem-
≈
thosefor2D-diffusionmodels. Interestingly,allthe three brane movement of only particles of proper valence at
2D-diffusionI(u0)-curveswithDxdifferentonthebehav- corresponding temperatures. Calculated current-voltage
ior or the value show very similar forms with the same characteristics also show that the channel model is non-
resonant potential um = 3.5. In addition, the isotropic Ohmic. The full characteristics of conductance exhibit
2D-diffusion model, Dx = Dz, provides the highest cur- an absolute maximum at the same resonant channel po-
rent peak. tential as that identified in the currents.
It should be conclusively noted that all the results
presentedaboveareprincipally relatedto the considered
IV. CONCLUSIONS singleparticlemodel,neglectingalltheeffectsassociated
with the many-particle couplings, the particle size, and
the potential induced by particle itself. So, these results
We have considered an anisotropic 2D-diffusion of
might be served as an argument for further studies.
a charged molecule (particle) through a large biologi-
cal channel under an external voltage. Connecting the
Acknowledgments. V.L.N. gratefully acknowledges
two reservoirs with different particle concentrations, the
a generous hospitality from Institute for Bio-Medical
channel is modeled as a rigid cylinder characterized by
Physics in Hochiminh City, where this work has been
the three structure parameters: the radius, the length,
done.
andthesurfacedensityofthenegativechargesofchannel
interior-lining. These negative charges induce inside the
channel a potential that is uniquely determined by the
channel structure parameters and that critically affects
the transmembrane particle movement. The suggested
model is rather phenomenological so that the channel-
induced potential can be calculated exactly. Neverthe-
less, it serves well to gain an understanding of the phys-
ical mechanism of the channel-facilitated particle move-
ment. More detailed quantitative models are required
to describe concrete realistic biological channels (see for
example, [32]).
Ourstudyisconcentratedonshowingtheinfluencesof
this channel-induced potential and the external voltage
onthe typicaldynamicalcharacteristicsofparticles such
as the translocation probabilities, the average translo-
cation times, the net current, and the channel ion con-
9
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