non-equilibrium stochastic dynamics of open ion channels · 2013-11-25 · non-equilibrium...

Nonlinear Phenomena in Complex Systems, vol. 16, no. 2 (2013), pp. 146 - 161

Non-equilibrium Stochastic Dynamics of Open Ion

Channels

R. Tindjong and I. KaufmanDepartment of Physics, Lancaster University, Lancaster LA1 4YB, UK

D. G. Luchinsky∗

Mission Critical Technologies Inc., 2041 Rosecrans Ave. Suite 225 El Segundo, CA 90245, USA

P. V. E. McClintock†

Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom

I. KhovanovSchool of Engineering, University of Warwick, Coventry CV4 7AL, UK

R. S. EisenbergDepartment of Molecular Biophysics and Physiology,

Rush Medical College, 1750 West Harrison, Chicago, IL 60612, USA

(Received 15 February, 2013)

We present and discuss a modified version of reaction rate theory (RRT) to describe thepassage of a positive ion through a biological ion channel. It takes explicit account of thenon-equilibrium nature of the permeation process. Unlike traditional RRT, it allows for thenon-constant transition rates that arise naturally in an archetypal model of an ion channel.In particular, we allow for the fact that the average escape time of an ion trapped at theselectivity filter (SF) can be reduced substantially by the pair correlations between ions:the arrival of a second ion at the channel entrance significantly reduces the potential barrierimpeding the escape of the ion from the SF. The effects of this rate modulation on the current-voltage and current-concentration characteristics of the channel are studied parametrically.Stochastic amplification of the channel conductivity by charge fluctuations is demonstratedand compared with the results of Brownian dynamics simulations.

PACS numbers: 02.50.Tt, 05.45.Tp, 05.10.Gg, 05.45.Xt

Keywords: ion channels, permeation, nonequilibrium rate, stochastic dynamics, fluctuating barrier

1. Introduction

Ion channels are pathways down the centersof proteins embedded into the membranes of bi-ological cells [1, 2]. Their purpose is to facilitatethe diffusion of ions across the cell membranes.To support life, channels have to be precise, oftenselecting between similar ions with an accuracy of1:1000 [3], yet still able to deliver millions of ions

∗Also at Department of Physics, Lancaster University,

Lancaster LA1 4YB, UK†E-mail: [email protected]

per second, i.e. almost at the rate of free diffu-sion. In recent years impressive progress has beenachieved both in understanding the structure ofion channels [4] and in modeling their proper-ties [5]. Yet the mechanism that enables channelsto be highly selective between alike ions, whilestill passing millions of ions per second, remainsa tantalizing paradox.

Reaction rate theory is one of the oldest andmost widely accepted theoretical tools used to ac-count for the operation of ion channels. It canreproduce some important channel properties, in-cluding their conductivity and their selectivity be-

146

Non-equilibrium Stochastic Dynamics of Open Ion Channels 147

tween alike ions. It treats the channels as a set ofstates, with fixed transition rates between them.The current consensus views the transition rates,and often the states themselves, merely as con-venient fitting parameters of the model withouthaving any direct physical meaning. The rationalebehind this view is the complexity of real channeldynamics, in which transition rates are modulat-ed by many different factors, including long rangeinteractions with ions in the electrolytes, interac-tions with fixed charge on the channel wall, theeffects of hydration and dehydration of ions with-in the channel, and wall vibrations. Given thissituation, equilibrium reaction rate theory (RRT)with a large number of states and fixed transitionrates between them, serves as a first order approx-imation to the complex non-equilibrium reality.

At the same, the validity of the assump-tion of constant transition rates, and their lackof physical significance, have often been ques-tioned [6]. It was shown for example that theuse of constant rates is inconsistent with exper-iment [7], and that the modulation of transitionrates due to random fluctuations in the electricpotential can be used to extract useful work fromthe energy of non-equilibrium fluctuations [8], e.g.to transport molecules in molecular motors [9].

We seek to account for, and understand, thenon-equilibrium stochastic phenomena associat-ed with ionic permeation though an open ionchannel. A possible theoretical approach is basedon an extension of RRT that takes the above-mentioned interactions into account explicitly,using the standard results of stochastic theory.Within this framework it may become possible toreconcile the parameters of the RRT model withthe actual properties of the channels and, in par-ticular, with the shape of the channel potentialsas obtained from molecular dynamics (MD) simu-lations. These latter can then be used in Browniandynamics (BD) simulations.

In this paper we review briefly some recentresults derived by this approach and consider itsapplication to the analysis of conductivity in anarchetypal open ion channel. We show that thetransition rate is modulated by long range pair

correlations of the ionic motion, and that thismodulation may result in stochastic amplifica-tion of the channel conductivity. The predictedstochastic amplification is compared with the re-sults of BD simulations.

The paper is organized as follows. Wepresent in Section 2 a modified archetypal modelof the channel. Section 3 develops a simple gener-alization of RRT that takes into account the ion-ion interactions. Parametric studies of the mod-el are described in Section 4. Predictions basedon it are compared with the characteristics of re-al channels, and with the results of BD simula-tions, in Section 5, where it is shown that non-equilibrium interactions can amplify the currentthrough the channel. Finally, we summarise themain results, draw conclusions, and outline ourperception of future perspectives in Section 6.

2. Model

We will base our simple extension of RRTon a generic model [1, 10, 11] of an ion chan-nel, consisting of a water-filled cylindrical holethrough the protein in the cell membrane, witha single binding site around its centre. The chan-nel is open to the bulk solutions on both sidesof the membrane, as sketched in Figure 1(a). Instandard RRT the channel is modeled as threepotential barriers with prescribed transition ratesin each direction over their tops, and the ratesare treated as fitting parameters in applying themodel to any specific kind of channel. To extendthis approach by including explicitly the modula-tion of the rates due to ion-ion interactions, wemodify the model in two important respects.

First, we introduce two effective bindingsites in the form of hemispheres at each side ofthe channel as shown in Figure 1(a). The physicalsignificance of these sites is related to the Debyescreening of the ionic potential in the electrolytesolutions. Thus, for a typical screening length λD

of order 5 A, only ions coming within λD of thechannel entrance can influence the motion of anion within the channel. This influence can be quite

Nonlinear Phenomena in Complex Systems Vol. 16, no. 2, 2013

148 R. Tindjong et al.

FIG. 1. (Color online) (a) Geometry of the model sys-tem under consideration (see text). (b) The total po-tential energy is made up from a sum of several differ-ent contributions. The curves represent (from top tobottom at x = 0): the self-potential; the potential dueto the externally applied voltage; the total potentialenergy; and the potential due to the fixed charge. Theends of the channel are at ±15 A as indicated by thevertical alignment with the schematic of the model in(a).

significant because the channel acts as an elec-trostatic amplifier of the ion-ion interaction. Thelatter phenomenon can readily be understood byconsideration of two limiting cases. For very largechannel radii the effective dielectric constant εeff

of the channel approaches its value in water, sothat the interaction potential φcorr ∝ 1/εwr de-cays fast. In the opposite limit of small channelradius (R → 0), εeff approaches its value in theprotein, and φcorr ∝ 1/εpr then decays up to 40times more slowly. In narrow channels the εeff

is expected to be close to the protein value [12],giving rise to a strong amplification of the ion-ioninteraction.

Secondly, to characterize the ion-ion interac-tion we introduce the following physical param-

−50 −30 −15 0 15 30 50−15

−10

−5

0

5

10

x (A)

U (

k BT

)

ω21

ω12

FIG. 2. (Color online) Potential energy along the chan-nel axis, for 200mV applied voltage. The full black lineis the potential of a single Na+ ion moving along thechannel axis. The dashed line is the potential actingon a Na+ ion moving along the channel axis whenthere is one Na+ ion at the mouth of the channel. Thevertical dash-dotted lines indicate the positions of theends of the channel. ω21 is the average inverse timeduring which the potential remains deep (full curve)before a Na+ ion arrives at the left entrance, makingthe potential shallower (dashed); ω12 is the average in-verse time during which the potential remains shallow,before the Na+ ion diffuses away from the channel en-trance, thereby returning the potential to its originaldeep configuration.

eters. The rate at which ions arrive at the leftchannel mouth is given by the Smoluchowsky rate1/τar ∝ πRCLD, where CL is the concentrationof ions in the left bath and D is the ionic diffusionconstant. The rate at which ions leave the hemi-sphere is given by the diffusion time τD ∝ R2/3D.These estimates were confirmed in our earlier 3DBD simulations [13, 14], where it was shown thatthe distribution of random arrival times follows anexponential law with the characteristic time givenby the inverse of the Smoluchowsky rate. In ad-dition, the parameter characterizing the strengthof the interaction is introduced as a change inthe depth of the exit potential barrier for an iontrapped at the binding site when the second ionarrives at the channel mouth.

The change in the potential barrier at thechannel exit can be estimated by taking into ac-

Нелинейные явления в сложных системах Т. 16, № 2, 2013


count the almost linear decay of the potential in-side the channel [15]. For example, if the secondion is located at a distance 1 A from the chan-nel mouth and the local potential extrema at thebinding site are separated by 7 A, the potentialbarrier height is reduced by ∆U2 ≈ 1.5 kBT andthe pair correlations exert a leading order effect onion motion through the channel. These estimatescan be further confirmed by explicit solution ofthe Poisson equation for the single-ion potentialprofile in the channel both with, and without, thesecond ion at the channel mouth.

Solutions of the Poisson equation for a singleion are shown in Figure 1 (bottom). The corre-sponding potential profile consists of the follow-ing components: the self-energy due to the imagecharges induced in the protein walls, the exter-nal potential corresponding to the applied poten-tial difference across the cell membrane (i.e. be-tween the two baths), and the potential of thefixed charge on the protein wall. The resultanttotal potential is show by the full blue line inFigure 1 (bottom). It exhibits a relatively deeppotential minimum at the center of the channel,corresponding to the binding site, and two poten-tial barriers impeding exit to the right and to theleft respectively.

With a second ion present at the left channelmouth, there is an additional contribution due tothe ion-ion interaction which modifies the singleion potential as shown by the dashed curve inFigure 2. It is evident that the modified potentialhas a lower barrier impeding exit to the right.The change Fcorr in the height of the potentialbarrier, found by solution of the Poisson equation,is of the order of 1.5 kBT in agreement with theabove estimates. For escape processes that occuron the timescale of a few tens of nanoseconds,only the averaged position of the second ion atthe channel mouth has physical significance. Thecorresponding fluctuations of U(x), averaged intime, can be modeled as dichotomous noise.

Note that, without having introduced anyfitting parameters into the model, we arrive natu-rally at a situation where the single-ion potentialof the channel is flipping between two states. The

flipping rate to the state with decreased potentialbarrier (state 2) is given by the Smoluchowskyarrival rate ω21 = J l

ar, and the rate of return tothe single-ion state with the larger potential bar-rier (ground state 1) is given by the diffusion rateω12 = kl

ref . The two different potential barrierconfigurations result in two different escape ratesof the trapped ion out of the channel. In order todescribe this situation we introduce in Section 3 amodified form of RRT that takes into account thenon-equilibrium modulation of the escape rate.

3. Non-equilibrium rate theory

The model introduced above has three sites:the binding site at the center of the channel; andan effective site at each of the channel mouths.The rate k2 at which ions escape to the right fromthe binding site is a function of the model parame-ters that characterize the interaction between theion at the binding site and ions arriving at ran-dom at the left channel mouth. We will develop atheory of the stochastic transition of ions throughthe channel below, in two stages.

First, we neglect the ion-ion interactions andapply standard RRT in Section 3 3.1 to calcu-late the current J through the channel using con-stant reaction rates, arriving at the simple formu-la J = k2P0. Secondly, in Section 3 3.2, we includethe interactions and calculate the dependence ofk2 on the model parameters ω12, ω21, and ∆U2.The results of these calculations are investigatedparametrically and compared with the results ofBD simulations in Section 4.



3.1. Reaction Rate Theory for ion chan-

nels conduction

The kinetic equations for the model present-ed in Sec. 2 are given by RRT (cf. [10]):

P0 = k1Pl(1 − P0) − (k−1 + k2)P0

+k−2Pr(1 − P0),

Pl = J lar − kl

refPl − k1Pl(1 − P0) + k−1P0,

Pr = Jrar − kr

difPr + k2P0 − k−2Pr(1 − P0).

(1)

Here J l,rar are the constant arrival rates at which

ions are injected respectively into the left andright volumes located at the mouths of the chan-nel. Pl,r are respectively the occupation probabil-ities of the left and right volumes. P0 is the prob-ability of occupation of the charged binding sitein the middle of the channel. Pl,r(1− P0) ensuresthat an ion can only enter an empty channel fromthe left or the right mouth. kl

ref and krdif are re-

spectively the diffusive reflection rates at the leftand right mouths of the channel. The current fromthe left to the right, or from the right to the left,mouth of the channel is given by:

J = k1Pl(1 − P0) − k−1P0

= k2P0 − k−2Pr(1 − P0). (2)

Looking for the steady state current, we write:

J lar − kl

refPl − J = 0,

k1Pl(1 − P0) − (k−1 + k2)P0

+k−2Pr(1 − P0) = 0,

Jrar − kr

difPr + J = 0.

(3)

The ion injection rate on the left is given by:J l

ar = 2πRCLD. When the channel is closed, theoccupation probability of the left hemisphere atthe channel mouth is given by Pl = 2πR3CL,where R is the radius of the hemisphere. The ioninjection rate and the occupation probability fromthe right mouth are written similarly. The diffu-sive reflection rates at the left and right mouthskl

ref and krdif are calculated from the station-

ary solution when the channel is non-conducting

(J = 0); this leads to

klref =

3D

R2, (4)

krdif =

3D

R2. (5)

Combining the elements of the steady state equa-tion leads to a second order equation in P0. Thecurrent can then be calculated as:

J =k1(1 − P0)J

lar − kl

refk−1P0

klref + k1(1 − P0)

. (6)

The occupation probabilities of the left and rightmouths of the channel are given respectively by:

Pl =J l

ar − J

klref

, (7)

Pr =J

krdif

. (8)

We are interested in the particular case of unidi-rectional current. The concentration on the righthand side of the channel is therefore set to ze-ro: CR = 0. Since there is no backflow from theright bulk to the left bulk, we set k−2 = 0. Forsmall values of the k2 parameter, correspondingto J ≪ J l

ar, the channel binding site occupancyis given by:

P0 =k1J

lar

k1J lar + (k2 + k−1)kl

ref

, (9)

and the corresponding current is given by:

J = k2P0. (10)

Note that the current through the channel in thisstrongly asymmetric situation depends only onthe two variables k2 and P0. In standard rate the-ory k2 is constant and the current is then expectedto follow the well-known Michaelis-Menten kinet-ics [1, 16], exhibiting saturation of P0 as a functionof J l

ar. However, if the interactions between ionscannot be neglected, the escape rate k2 becomesa function of J l

ar and klref . As a result the cur-

rent through the channel may deviate markedly



from the Michaelis-Menten form, demonstratingstochastic amplification.

To seek further insight into the influence ofthe ion-ion interaction on the permeation of thechannel, we now introduce in Section 3 3.2 a non-equilibrium extension of RRT by calculation ofthe dependence of k2 on the interaction parame-ters.

3.2. Fluctuating Barrier Theory

To take account of ion-ion interactions, andto calculate the escape rate k2 as a function of theflipping rates and the corresponding modulationof the potential barrier k2 = k2 (ω12, ω21, ∆U2),

we notice that the modulation of the potential isstrongly non-equilibrium. This is because at leastone rate ω12 is of the order of the inverse diffusiontime, which is close to the relaxation time of theion to the potential minimum.

Thus, k2 has to be calculated usingChapman-Kolmogorov theory [17] for mixeddiffusion-jump processes. Accordingly, we seekthe probability density Pi(x, t|y, t′ = 0) of thestochastic variable, starting at t′ = 0 at positiony, to exit the channel either at x1 or x2, while thesystem displays mixed dynamics, jumping withthe rate ω(j|i, t′) between the potential states iand j that have drift and diffusion coefficients Ai,j

and Bi,j respectively:

∂t′Pi(x, t|y, t′) = −Ai(y, t′)∂yPi(x, t|y, t′) − 1

2Bi(y, t′)∂2

yPi(x, t|y, t′) (11)

+∑

j

ω(j|i, t′)[Pj(x, t|y, t′) − Pi(x, t|y, t′)].

Following Gardiner [17] we seek the probabili-ty density Pi(x, t|y, 0) for the stochastic variable,starting at t′ = 0 at position y, to exit the channeleither at the left or the right barrier top. Usingthe standard formalism [17] the mean value of theexit time, the so-called mean first passage time(MFPT) τ(y), can be expressed via Pi(x, t|y, 0)as

τ(y) =

∫ ∞

0

∫ x1

x0

Pi(x, t|y, 0)dxdt, (12)

where x0 is the potential minimum and x1 is thepotential maximum close to the right mouth.

To estimate the escape rate we analyze thecase of the two-state potential both without, andwith, one positive ion at the left channel mouth(as shown in Figure 2), setting the ionic concen-tration in the right bulk solution to zero (CR = 0).In this case the solution of the Eq. (11) can bereduced [18, 19] to the analysis of the following

system of equations [20]

τ ′1 = S1

kBT

mγS′

1 =1

mγφ′

1S1 + ω21(τ1 − τ2) − 1

τ ′2 = S2 (13)

kBT

mγS′

2 =1

mγφ′

2S2 + ω12(τ2 − τ1) − 1,

supplied with boundary conditions τ ′i(x =

x0, t) = 0; τi(x = x1, t) = 0. The escape ratek2 = 1/τ1 can now be found as a solution of theboundary value problem (13).

We are now in a position to investigate non-equilibrium phenomena associated with channelpermeation for different sets of parameters.

4. Parametric study of the model

As discussed above, the escape rate k2 is afunction of several parameters and, in particular:



the ions’ arrival rate ω21 = J lar; their diffusion

rate ω12 = klref ; and the strength of their mutual

interaction ∆U2. We now consider the effect ofchanges in each of these quantities.

4.1. Dependence on the flipping rates

To consider the dependence of the escaperate on the flipping rates we note that the w21 =J l

ar ∝ πRCLD, so that the dependence on w21

can be interpreted in terms of concentration CL

in the left bulk solution. On the other hand theflipping-back rate w12 = kl

ref does not dependon the concentration (but, rather, corresponds tohow long the ion at the left mouth takes to dif-fuse away). In real channels, however, the poten-tial profile will depend on the precise geometryand on possible additional small fixed charge atthe channel mouth. It is therefore of interest toanalyze the dependences of the escape rate andcurrent as functions of both w21 and w12.

Note that an increasing concentration leadsto an increasing w21 and more frequent modula-tion of the unperturbed potential. Since the mod-ified potential barrier is reduced by an increment∆U2, it is expected that the more frequent arrivalof ions at the channel mouth may result in a sig-nificant increase of the escape rate k2 and thus ofthe channel current.

On the other hand, an increasing w12 meansthat after arrival the ion will spend less time atthe channel mouth, and hence that the potentialbarrier will spend more time in its unperturbedstate. It is therefore expected that the escape ratek2 and the channel current will both be decreasingfunctions of the w12.

Calculations of the k2 as a function of w21 =J l

ar ∝ πRCLD are presented in Figure 3 for differ-ent values of w12. It can be seen that, as expected,the escape rate k2 is an increasing function of theconcentration and a decreasing function of w12.

Another interesting feature evident in thefigure is that the k2 − C curves take thewell-known sigmoid shape also known as Hill’scurve [21, 22]. For example Hill’s curve was found

100

101

102

103

104

105

0

2

4

6

8

10

12

14x 10

6

CL (mM)

k 2 (s−

1 )

0.02ω12

0.04ω12

0.1ω12

0.2ω12

0.4ω12

1ω12

FIG. 3. Escape rate k2 as a function of the left-bath concentration CL and reflection diffusion rateω12 (shown in key) of ions at the left mouth. Thepotential difference applied across the channel wasVapp = 200mV.

in the calcium-activated potassium channel [23],in which an additional binding site exists for Caions. Thus our theory offers an alternative deriva-tion of Hill’s curve based on the fundamental elec-trostatic properties of the channel.

To obtain the current we note that, as shownabove in Sec. 3, J = k2P0, where the channel pop-ulation P0 is given by Equation (9), which is anonlinear function of both Jar ∝ CL and k2. Thechannel population and the resulting current cal-culated within formalism of non-linear RRT areshown in Figure 4.

The main result of this analysis is the pre-diction of a strong increase of the current throughthe channel as a function of both ionic concen-tration and ionic residence time at the channelmouth. Note that there is an important differencebetween this picture and the standard knock-onmechanism, and between their predictions. Here,the second ion does not, on average, enter thechannel at all.

The results of Figure 4 corroborate this pic-ture by showing that the current increase cor-responds to the plateau region where the chan-nel population remains at saturation level, witha single ion occupying the channel nearly all thetime. This fact is very important from the point ofview of applications. Consider for example a situ-



10010

110210

310410

5

02

46

810

0

1

2

3

ω12

(×108 s−1)C

L (mM)

I (pA

)

10010

110210

310410

5

02

46

810

0.4

0.6

0.8

1

ω12

(×108 s−1)C

L (mM)

P 0

(b)

(a)

FIG. 4. (Color online) (a) The channel population as afunction of the left-bath concentration CL and the av-erage inverse diffusion time w12 of the second Na+ ionaway from the channel entrance. Here, ∆U2 ≈ 1kBTand the potential difference applied across the channelwas Vapp = 200mV. (b) The corresponding permeatingcurrent.

ation when the conducting ions are mixed withanother type of ion in the solution in the ra-tio 1:20. The mechanism proposed above suggeststhat the more-abundant ions will arrive at thechannel mouth at a rate 20 larger than that of theconducting ions. Upon arrival they will stronglymodulate the single-ion potential for the conduct-ing ion occupying the channel, thereby facilitatingcompletion of the permeation process.

We conjecture that the proposed mechanismcan he used to extract useful work from the non-equilibrium modulation of the fluctuating channelpotential due to the frequent arrival of the more-abundant non-conducting ions. For example thisenergy can switch the channel between differentconformational or dynamical states.

4.2. Dependence on the strength of the

interaction

We now turn to an analysis of stochastic am-plification of the channel current, as a function ofthe interaction strength. As discussed in Sec. 2,the strength of the ion-ion interaction in this sys-tem can be characterized by the change ∆U2 inthe potential barrier height for ions exiting to theright from the binding site.

Within our electrostatic model of channelconductance, the strength of the potential mod-ulation is characterized by: (i) the locations ofthe potential extrema; (ii) the distance betweenthe potential minimum and the right-hand max-imum; and (iii) the effective averaged position ofthe second ion at the left channel entrance. Thefirst two factors are determined mainly by the dis-tribution of fixed charge on the channel wall. Inthe present case we consider the latter distribu-tion to be constant, and our analysis is thereforerestricted to the dependence of the exit rate onthe effective location of the second ion near thechannel mouth.

In real channels the location of the secondion can be affected by various factors includingthe precise geometry and the possible presence ofweak additional charge at the channel mouth. Theeffective location of the second ion at the channelmouth can also be a function of the concentration.Indeed, when the concentration increases the De-bye screening length decreases as ∝ 1/

√CL.

Even small variations in the effective lo-cation of the second ion may result in markedchanges in the strength of the interaction. This isbecause the ion’s potential decays as ∝ 1/d, whered is the distance between the ion and the channelentrance plane. Because of this singular behaviorwe expect that even small (of order 0.5A) changesin the effective ion location will result in signifi-cant changes in the modulation of the potentialdue to the ion-ion interaction. Furthermore, evensmall changes in the amplitude of modulation willlead to substantial changes in the current throughthe channel.

To illustrate this point we consider the de-



101

102

103

1

2

3

4

5

6

7

8

9

10

x 107

CL (mM)

k 2 (s−

1 )

0.91∆U2

0.92∆U2

0.93∆U2

0.95∆U2

0.99∆U2

FIG. 5. Escape rate as a function of concentration fordifferent values of channel’s exit barrier height, rang-ing from 0.91∆U2 and 0.99∆U2, as indicated in thekey. The potential difference applied across the chan-nel was Vapp = 200mV.

pendence of the escape rate k2 on the amplitudeof the channels’ exit barrier modulation ∆U2, forvarious concentrations of ions in the left bulk so-lution. We restrict the changes in the potentialmodulation to small values of a few % between0.91∆U2 and 0.99∆U2. The results of the corre-sponding calculations are plotted in Figure 5.

It can be seen from the figure that, in agree-ment with the above considerations, above evena very small (less than 1%) increase in the mod-ulation depth of the ∆U2 results in a more thanfour-fold increase of the exit rate and thus of thechannel current.

To conclude our discussion of parametricstudies, we now consider the current-voltage char-acteristics of the model for different concentra-tions of ions in the left bulk.

4.3. Current as a function of the applied

voltage and concentration

In considering the current-voltage character-istics, we recall that the contribution of the ex-ternally applied voltage to the total channel po-

0 0.05 0.1 0.15 0.20

1

2

3

4

5

6

7x 10

6

Vapp

(V)

k 2 (s−

1 )

CL = 0.002M

CL = 0.003M

CL = 0.01M

CL = 0.1M

CL = 0.5M

CL = 1M

FIG. 6. The escape rate k2 as a function of the poten-tial difference Vapp applied across the channel, for theleft-bath concentrations shown in the key.

tential (see Figure 1) is a linearly-decaying func-tion. As a result the potential barrier at the chan-nel exit decreases linearly with applied voltage.For example the height of the potential barrier isdecreased by ≈ 2kBT when an external voltageVapp = 200mV is applied across the channel.

If the ion-ion interaction is not taken intoaccount, the exit rate for an ion starting from thebottom of the potential xa to reach the top ofthe potential barrier xb can be estimated usingKramer’s formula

rK =D

√

U ′′(xa)|U ′′(xb)|2πkBT

× exp

(

−(U(xa) − U(xb))

kBT

)

. (14)

It is thus expected to be an exponential functionof applied voltage, and not to depend on the con-centration. In this case, according to Eq. (10),the current across the channel will depend on theconcentration only via the channel population P0.This latter dependence is a saturating function ofconcentration of the Michaelis-Menten type.

When the ion-ion interaction is present itwill reveal itself in two respects. First, the escaperate will become an increasing function of the con-centration due to the increasing rate of modula-tion of the escape potential. Secondly, the relative



depth of the potential barrier modulation will in-crease with the applied voltage, because the ab-solute value of the barrier will decrease, whereasthe change of the potential due to the ion-ion in-teraction does not depend on the applied voltage.In other words, we expect a super-exponential de-pendence of the escape rate k2 on the value of theapplied voltage.

Both of these characteristic features of theion-ion interaction can be seen in the calculateddependence of the escape rate on the applied volt-age, for different values of the concentration, asshown in the Figure 6. The dependence of the es-cape rate on the concentration can be seen in thefigure as an overall upwards shift in the k2(Vapp)curves as the concentration is increased.

Finally, we note that in the absence of ion-ion interaction the channel current was expectedto be independent of the concentration in the re-

10010

110210

310410

5

00.05

0.10.15

0.2

0

1

2

3

Vapp

(V)C

L (mM)

I (p

A)

10010

110210

310410

5

00.05

0.10.15

0.2

0

0.2

0.4

0.6

0.8

1

Vapp

(V)C

L (mM)

P0

(b)

(a)

FIG. 7. (Color online) The permeating current as afunction of the left-bath concentration CL and the po-tential difference Vapp applied across the channel.

gion of saturated channel population, while non-equilibrium RRT predicts strong increase of thecurrent in this region. The results of these lat-ter predictions are shown in Figure 7. The strongincrease of the current in the region where popu-lation P0 is saturated is clearly evident.

To verify the predicted features of the ionchannel current in the presence of the ion-ion in-teraction we performed BD simulations, as de-scribed below.

5. Comparison with BD simula-

tions

5.1. Model Equations

The BD simulations of the model describedin Sec. 2 and shown in the Figure 2 are based onthe integration of the Langevin equations (15) forinteracting ions moving along the channel axis,coupled to simultaneous solution of the Poissonequation (16) for the channel, considering bothfixed and mobile charges in an axisymmetric ge-ometry.

The motion of interacting ions of mass m andcharge q through the channel is modeled using theoverdamped coupled Langevin equations

mγdxi

dt= qEi + Fi,corr +

√

2mγkBTξi(t). (15)

The terms on the left-hand side representCoulomb, frictional, and random forces respec-tively. The latter is a zero-mean, Gaussian, ran-dom force due to the fluctuating environment. It isrelated to the frictional force via the fluctuation-dissipation theorem [24, 25]. E denotes the electricfield at the ion’s position due to the fixed chargeof the protein, the membrane potential, and thesurface charge induced at the protein-water inter-face.

The term Fcorr corresponds to the interac-tion of the ion at the binding site with othermobile ions. This term is always present in BDsimulations (see e.g. [25]). However, in the tran-sition from BD simulations to further theoretical



approximations, such as RRT [10] or the Poisson–Nernst–Plank (PNP) equation [24, 26], the effectof this interaction on the escape rate from thebinding site is usually neglected.

The Poisson equation is given by:

∇ · [ε(r)∇φ(r)] = −[

N∑

i=1

qjnj(r) + pfx(r)

]

(16)

where qj is the charge of the j-th ion and pfx isthe fixed charge distribution on the channel wall.The channel is modeled as a cylindrical tube oflength L and radius R with a fixed charged ringat its middle, as illustrated in Figure 1.

5.2. Coupled solution of Langevin and

Poisson equation using a lookup table

To solve Equation (16) we use a finite-volume method discretized on a staggered rectan-gular grid. The solver is able to accommodate thesevere jumps in dielectric permittivity typical ofion channels (εw = 80 and εp = 2 respectively forwater and protein) and allows one to calculate theelectrostatic potentials (see Figure 2) and forcesin ion channels, including the image forces act-ing on ions near the water-protein boundary, theself-potential arising from the membrane protein,the contribution to the potential due to the fixedcharge located at the channels’ binding site, andthe contribution of the external voltage appliedacross the channel.

The BD simulation involves solving Poissonequation for each time step of ions in Langevinequation. This process is very time-consuming es-pecially in the bulk where ions spend most of thetime during the simulation. To facilitate the nu-merical integration, the potential energy for a sin-gle ion was stored in a table for all locations of theion along the channel axis. To calculate the fieldfor an arbitrary number of ions in the system, theprinciple of superposition was used.

5.3. Charge fluctuations and injection

Since our primary target is to calculate ionconduction due to ions crossing the channel afurther reduction of the simulation time is pos-sible by avoiding the time-consuming simulationof the bulk solution. We have shown numericallyin our earlier work that such a reduction is pos-sible because ionic arrivals at the channel mouthfrom the bulk solution is governed by the Smolu-chowski arrival rate [14] at the channel mouth.For this reason, simulations in the bulk solutioncould be neglected by, instead, injecting ions us-ing the Smoluchowski arrival rate near the chan-nel entrance and allowed them to diffuse on theaxis. During this process, the majority of the ionsin the simulations diffused away from the chan-nel; occasionally a single ion entered the channeland continue to fluctuate near the SF binding site.Eventually this ion exited the channel by hoppingover the exit potential barrier. The correspondingion current was calculated by counting the num-ber of ions passing the channel per unit of time.Further details of the simulation and the valida-tion of the algorithm can be found in e.g. [14].

5.4. Comparison with gA channel

Before illustrating the phenomenon ofstochastic amplification in the BD simulations,we first demonstrate that our proposed non-equilibrium version of RRT is indeed an exten-sion of standard RRT, and can reproduce its re-sults for corresponding values of the model pa-rameters. ln doing so we also wish to demon-strate that non-equilibrium RRT is capable of re-producing the experimentally measured current-concentration (I − C) characteristics of real ionchannels. To this end we compare the model pre-dictions with the experimentally measured I −Ccharacteristic of the gA channel [20].

The results of the comparison are shown inFigure8. It can be seen that, for the parame-ters of this channel, the theory is in reasonableagreement with the experimental data, as well



0 0.5 1 1.5 20

1

2

3

4

I (pA

)

CL (M)

ExperimentRR BD

FIG. 8. (Color online) Permeating current I of Na+

ions in the gA channel, as a function of left-bath con-centration CL, with a potential difference of 200mVapplied across the channel. Experimental data fromAndersen et al. [27] are plotted as circles. The lozengesrepresent calculations based on the modified rate mod-el; and the stars are from the BD simulations.

as with the BD simulations, and that it demon-strates standard Michaelis-Menten kinetics.

We now apply BD simulations to verify thepossibility of stochastic amplification predicted inthe current research.

5.5. Stochastic amplification of the chan-

nel current

We have performed BD simulations of thecurrent through the open ion channel for the se-lected model parameters: length L = 30A; ra-dius R = 4A; and fixed charge at the bind-ing site −0.81e. BD simulations of the current-concentration dependence for an applied voltageof 200mV are shown by the blue circles in Fig-ure 9. These results dependence exhibit satura-tion, with standard near Michaelis-Menten kinet-ics, up to a concentration CL ≈ 200 mM. Forhigher concentrations a strong stochastic ampli-fication of the channel current can be observed.The BD simulation results are compared with

those from non-equilibrium RRT (see Sec. 4 4.3)as shown by the grey surface. The deviations ofthe current from Michaelis-Menten kinetics above200 mM were predicted by the parametric stud-ies discussed in Sec. 4 4.1 and illustrated in theFigure 3.

However, an unexpected additional stochas-tic amplification of the channel current was ob-served in the BD simulations. It can be seen inthe figure as a deviation of the BD results (bluecircles) from the red line corresponding to thepredictions of non-equilibrium RRT. We attributethis additional amplification to the shift with in-creasing bulk concentration in the effective (av-eraged) position of the 2nd ion at the channelmouth. This effect can readily be understood onthe basis of the parametric studies in Sec. 4 4.2,where it was shown that a small shift in the ion’slocation near the channel mouth results in a smallchange in depth of the potential modulation ∆U2,leading in turn to a marked increase in current.

We are able to observe the correspondingsmall shift in the averaged ion location in the BDsimulations. This shift is clearly seen e.g. in theconditional PDF shown in the Figure 10 (see be-low).

The prediction of our modified RRT, takingaccount of small shift in the location of the sec-ond ion at the left channel, is shown by the dashedblue line in Figure 9: the agreement is clearly verygood, and we conclude that theory is then able todescribe accurately all the characteristic featuresof the current-concentration dependence that ap-pear in the BD simulations.

We now discuss in detail how the proposedmechanism of stochastic current amplification dif-fers from the well-known knock-on mechanism ofchannel conductance, and illustrate how this dif-ference can clearly be detected in the BD simula-tions.



50100

150200

250 100

101

102

103

1040

0.5

1

1.5

2

CL (mM)V

app (mV)

I (pA

)

FIG. 9. (Color online) The permeating current I as afunction of the left-bath concentration CL and appliedvoltage Vapp, calculated by modified RRT (the greysurface). The results of BD simulations (blue circles)are shown with Vapp = 200mV for comparison; the redline is a smooth curve drawn through their projectiononto the surface, as a guide to the eye.

5.6. Conditional escape PDF and com-

parison with standard knock-on model

An equilibrium distribution function ρst(x)is usually employed to characterize the ion’s lo-cation in the system. It gives the time-averagedprobability of finding an ion at a given locationin the system. The corresponding distribution forour model is shown in Figure 10(a) for differentconcentrations. It is evident that the ion withinthe channel is sharply localized near the bindingsite at x = 0, while the ionic distribution outsidethe channel is broad and nearly uniform. It canalso be seen from the figure that the channel occu-pancy, i.e. the averaged probability of finding anion at any location within the channel, increaseswith concentration until it reaches its saturationvalue ≈ 1. Note also that at any concentrationthe total number of ions outside the channel inour simulations is . 1.

However, the time-averaged distribution ofions within the system does not allow one to char-

−50 −40 −30 −15 0 15 30 40 500

0.1

0.2

Occ

upan

cy

x (A)

−50 −40 −30 −15 0 15 30 40 500

50

100

150

ρ(x)

(A

U)

x (A)

(b)

(a)

FIG. 10. (Color online) (a) The occupancy as a func-tion of position x, for different values of the left-bath concentration CL, ranging from 0.002M (bottomcurve) to 4.5M (top). (b) The conditional probabilitydistribution for the position of the second ion at themoment when the first ion escapes over the exit bar-rier (see Figure 2, for the same set of concentrations.The vertical dashed lines indicate the positions of theends of the channel.

acterize the strength of the ion-ion interaction orits effect on the escape of ions from the bindingsite.

To measure ion-ion pair correlations in theBD simulations we have introduced a new quan-tity – the conditional probability distributionρ(x) = P (x2 = x|x1 > xex) that gives the proba-bility of finding a 2nd ion in the system at locationx at the time instant t when the 1st ion is exitingthe system, i.e. is located near the top of the exitpotential barrier xex. The conditional PDF ρ(x)is normalized by the total probability of findingions anywhere in the system.

The conditional probability introducedabove allows us to determine the location of thesecond ion during the escape and thus to discrim-inate between different escape mechanisms. lnparticular, we can distinguish clearly between thestandard knock-on mechanism (see e.g [28, 29])



and the mechanism for enhancement of thechannel current proposed here. The standardknock-on mechanism, proposed by Hodgkin andKeynes [30], assumes that the second ion entersthe channel, lowering the exit potential barrierfor the first ion; the first ion then leaves thesystem, while the second remains in the channel.Effectively, therefore, the second ion pushesthe first one out and takes its place at the SF.Some features specific for the standard knock-onmechanism are that: (i) it involves a step whenboth ions are in the channel; (ii) the second ionremains in the channel when the first leaves it;and (iii) for selective conduction both ions mustbe of the same type.

In contrast, the mechanism proposed heredoes not require the second ion to enter the chan-nel; neither does it require both ions to be of thesame type. In terms of conditional escape proba-bilities the two mechanisms can be clearly distin-guished as follows. Because the traditional knock-on mechanism requires that the second ion enterthe channel and remain there during the escapeby the first ion, the conditional PDF ρ(x) shoulddisplay a peak at the channel binding site, indi-cating that the 2nd ion is located at the bindingsite when the 1st ion is exiting the channel. In ourproposed mechanism, the 2nd ion can stay outsidethe channel during the whole escape process. Ac-cordingly, the conditional PDF ρ(x) should dis-play a peak at the channel entrance, indicatingthat the 2nd ion is as likely to be located outsideof the channel as inside it at the moment whenthe 1st ion is crossing the exit barrier.

Figure 10(b) exhibits a clearly resolved peakin the conditional PDF ρ(x) at the channel en-trance. It appears at CL ≈ 0.5M and is increasingas a function of concentration, indicating that ex-it over the potential barrier is strongly correlatedwith the presence of the 2nd ion at the channelmouth.

Interestingly, a transition to the standardknock-on mechanism is also clearly captured bymeasurements of the ρ(x). It can be seen fromthe figure that a second peak in the condition-al escape PDF is appearing at the binding site

(x = 0) for concentrations above 3M . It clearlyindicates that the standard knock-on mechanismis also possible, with the second ion to enteringand remaining in the channel during the escapeevent.

6. Discussion and conclusion

In this review we analyze the influence ofion-ion interaction on the rates at which ions per-meate open ion channels. Of special interest tous is the situation that arises when one ion isbound at the selectivity site while the second ionis freely diffusing outside the channel. This situ-ation is quite general, and we have shown thatit leads naturally to a strong modulation of thetransition rates. To take this modulation into ac-count we introduced a non-equilibrium extensionof reaction rate theory, and we performed para-metric studies of the channel characteristics onthis basis.

We found that the rate at which an ion ex-its the channel can be strongly enhanced by theconcentration, applied voltage, ion location at thechannel mouth, and diffusion time of the (exter-nal) ion away from the channel mouth. We ob-served a corresponding enhancement of the chan-nel current as a function of concentration in ourBrownian Dynamics simulations, and was showedthat it was in good agreement with the theory.

In addition, we found from the BD simula-tions that the mechanism by which the channelcurrent and be stochastic amplified is distinctlydifferent from the well-known knock-on mecha-nism, except at the highest concentrations. Todistinguish clearly between the two mechanismswe introduced a novel conditional escape PDF.We were thus able to establish that both mech-anisms are possible, that they correspond to twodifferent peaks in the conditional PDF, and thata transition occurs between the two mechanismswhen the concentration becomes high enough.

The distinguishing feature of stochastic am-plification is that, unlike conventional knock-on,the second ion does not on average enter the chan-



nel, which means that it can be of a different typefrom the conducting ions.

This distinction carries interesting implica-tions for the selectivity vs. conductivity paradox,partially resolving it. The paradox can briefly beformulated as follows: strong selectivity requiresdeep potential wells; but deep potential wells im-ply low conductivity. The proposed mechanismsresolves this contradiction in the following way.Because the ion arriving from the bath does notneed to enter the channel in order to reduce verysignificantly the escape time of the ion at the se-lectivity filter, conduction can be enhanced by thepresence of different species of ions. So if there aree.g. two types of cations, Na+ and K+ in the ra-tio 20:1, the very presence of the Na+ ion willenhance the conductivity of the K+ ions withoutthe Na+ ion needing to enter the channel.

In conclusion, we have studied the dynamicsof ions permeating an archetypal model ion chan-nel. We showed that a modulation of the transi-tion rates arises naturally in ion channels due tolong range pair-correlations of the ions. The re-sultant strong current enhancement occurs priorto the possibility of conventional knock-on takingplace place, i.e. the second ion does not have toenter the channel before the exit of the first oc-curs, so that it is likely to be more widespreadand universal.

Acknowledgements

The research was supported by the Engineer-ing and Physical Sciences Research Council UK(grant No. EP/G070660/1).

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