Eur. Phys. J. B (2018) 91: 10https://doi.org/10.1140/epjb/e2017-80569-5 THE EUROPEAN

PHYSICAL JOURNAL BRegular Article

Physical properties of voltage gated poresLaureano Ramırez-Piscina1,a and Jose M. Sancho2

1 Department de Fısica, Universitat Politecnica de Catalunya, Avinguda Doctor Maranon 44, 08028 Barcelona,Spain

2 Departament de Fısica de la Materia Condensada, Universitat de Barcelona, Universitat de Barcelona Instituteof Complex Systems (UBICS), Martı i Franques 1, 08028 Barcelona, Spain

Received 9 October 2017 / Received in final form 17 November 2017Published online 15 January 2018 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2018

Abstract. Experiments on single ionic channels have contributed to a large extent to our current view onthe function of cell membrane. In these experiments the main observables are the physical quantities: ionicconcentration, membrane electrostatic potential and ionic fluxes, all of them presenting large fluctuations.The classical theory of Goldman–Hodking–Katz assumes that an open channel can be well described by aphysical pore where ions follow statistical physics. Nevertheless real molecular channels are active pores withopen and close dynamical states. By skipping the molecular complexity of real channels, here we present theinternal structure and calibration of two active pore models. These models present a minimum set of degreesof freedom, specifically ion positions and gate states, which follow Langevin equations constructed from aunique potential energy functional and by using standard rules of statistical physics. Numerical simulationsof both models are implemented and the results show that they have dynamical properties very close tothose observed in experiments of Na and K molecular channels. In particular a significant effect of theexternal ion concentration on gating dynamics is predicted, which is consistent with previous experimentalobservations. This approach can be extended to other channel types with more specific phenomenology.

1 Introduction

Ionic transport across the cell membrane is a very commonprocess in all cells. In most situations the transport is doneby very specific pores called molecular channels. Thesedevices are embedded into lipid membranes and operatebetween different ionic concentrations at both sides, andsubjected to the membrane voltage. Molecular channelsare not passive pores but they have internal structures(gates) that present conformational states such as openand close configurations. When the channel is in the openstate it allows the flux of charges driven by the ionicdensity gradient and the membrane electrostatic poten-tial. This flux can modify also this potential. Moreoverthe whole process is very selective: only a particular ioncan cross a specific channel. In the close state no flux ofions is in principle allowed, except for some small leak.The transitions between these two conformational statesare controlled by the membrane potential and thermalfluctuations.

The synchronized dynamics of a large number of suchchannels of several types, coupled to the ionic concen-trations at both sides of the cellular membrane, providesmechanisms for action potentials in neurons, cardiac cells,etc. [1–3]. Beyond the study of the conductivity proper-ties of the membrane (i.e. resulting from a large numberof channels), experiments performed on single channels

a e-mail: laure.piscina@upc.edu

have provided a good deal of information on the gat-ing dynamics of individual channels [2]. Such experimentshave shown that, in addition to the inherent randomnessof the gate open-close dynamics, there are also very strongfluctuations in the charge flux.

The biochemical structure of a molecular channel isquite complex at the molecular level. From the physi-cal point of view the relevant observables of a channelare the ionic concentrations, the membrane electrostaticpotential, and the ionic fluxes. Here we will address thedynamics of these observables by modeling the channel asa simple active pore, with a reduced set of observables ful-filling known physical laws and consistently incorporatingthermal fluctuations.

Most theoretical modelings follow the Hodking–Huxleyapproach [4] for the dynamics of membrane permeabil-ity. Fluctuations have been incorporated by using eitherLangevin or master equations [5–9]. In previous publi-cations, a semi-microscopic Langevin modelization wasused to show the excitable properties of a single poremimicking a Na channel in the presence of a leak of Kions [10], and the periodic firing of a pair of Na, K-likeactive pores [11]. In this approach all microscopic rele-vant variables (a reduced set of degrees of freedom) followstochastic (Langevin) dynamical equations according tovery general principles of statistical physics. The main aimis to use only the necessary physical mechanisms involved,and to obtain their effects by using standard physicallaws applied in a consistent way. Additional biological

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complexity would indeed be necessary to explain morespecific physiological features, but it is here stripped awayin order to identify the essential elements for explain-ing the basic channel gating phenomenology observed inexperiments.

By using this approach we will study in detailthe stochastic dynamics of two different pore models.Although we will reduce the elements of the modeling tothe minimum necessary to account for the behavior of sin-gle channels, we will show that the dynamics do presentthe behavior observed in Na and K channel experiments.Moreover the physical consistency of the model results tobe a key element in such agreement.

The outline of this paper is as follows. In Section 2, wesummarize the theoretical structure of the approach. InSection 3, we present the calibration and dynamical prop-erties of two pore models, and its comparison with theavailable experimental information. Finally in Section 4,we end with some conclusions and perspectives. The spe-cific technical details of the approach are presented inAppendix A.

2 Gating pore models

2.1 Summary of the approach

Our approach [10] follows the path opened by theGoldman–Hodking–Katz (GHK) equation [12] and theHodking–Huxley theory [4]. It is formulated by construct-ing an energy functional modeling the interaction betweenall variables, over which the application of standard rulesof statistical physics leads to a dynamics described interms of Langevin equations. The basic constituents of themodel are the following: (i) the movement of ions can bemodeled with Langevin equations; (ii) the gate-ions inter-action appears as the form of a barrier potential, whoseheight depends on the state (open or close) of the gate;(iii) the dynamics of the gate is modeled by a variable fol-lowing a Langevin equation with a potential energy thatdepends on the membrane potential; and finally (iv) themembrane potential obeys the physics of a capacitor.

The mechanical variables of the model are the positionof the ions xi (inside the channel domain [0, L]) and thegate variables Yj , where j indicates the possibility of sev-eral gates. All the physical information concerning thesevariables are incorporated into a single potential:

U(xi, Yj ,∆V ) =∑i

Vi(xi,∆V ) +∑j

V (Yj ,∆V )

+∑i,j

VI(Yj , xi). (1)

Here the term Vi(xi,∆V ) (see Eq. (A.1)) is the potentialenergy originated by the membrane potential ∆V on theions inside the pore at position xi. This term is responsi-ble for the physics contained in the GHK equation. Theterm V (Yj ,∆V ) (see Eq. (A.2)) is the potential due tothe membrane potential for the dynamics of the j-gate,represented by the variable Yj . The last term VI(Yj , xi)

(see Eq. (A.3)) corresponds to the interaction of i-ionswith the j-gate. See explicit expressions of these termsin Appendix A. Note that we have neglected ion–ioninteractions inside the pore, according with the hypothe-ses behind the GHK equation. Such interactions couldstraightforwardly be implemented into the approach ifmore quantitative results were required.

According to this energy functional, the dynamics forthe physical variables xi, Yj is given by the following setof Langevin equations:

γxxi = −∂xiU(xi, Yj ,∆V ) + ξi(t), (2)

γYj Yj = −∂YjU(xi, Yj ,∆V ) + ξYj

(t), (3)

where thermal noises fulfill

〈ξa(t)ξb(t′)〉 = 2γa kBT δa,b δ(t− t′), (4)

and γa are the corresponding frictions.We model thus the motion of the ions inside the channel

as a one-dimensional brownian motion, driven by ther-mal fluctuations, electrostatic potential and the differentionic concentrations between the two sides of the mem-brane. Whereas fluctuations and potentials are explicitlyput into the Langevin equations, the ionic concentrationsappear as boundary conditions at both ends of the chan-nel. Regarding gating, we have assumed that the degree offreedom Yj behaves as a nonlinear spring with two steadystates: YC ∼ 0 (close) and YO ∼ 1 (open). This hypoth-esis is similar to the modeling of the gating currents ofreference [13], where gating experiments were correlatedto a model in which a gating variable undergoes Brownianmotion in a one-dimensional diffusion landscape.

These equations are complemented with the capacitorequation for the dynamics of the membrane potential

CMd∆V

dt=∑i

Ii, (5)

where CM is the membrane capacity assumed to be con-stant and the r.h.s term includes all the ionic fluxes eitheracross the pore, membrane leaks or from an externalperturbating flux.

The most important feature of this approach is that allinteractions between ions and gates come from the energyfunctional of equation (1), and hence statistical mechan-ics can be applied consistently. In particular the dynamicalequations verify fluctuation–dissipation relations and theparameters of the model have a clear physical meaning.Statistical physics consistency is most relevant, since themain features of the results will not depend on parame-ter fits. The system is autonomous, and the only sourceof energy, apart from an applied electrostatic potential,is the chemical energy associated with the different ionicconcentration at both sides of the membrane.

Furthermore the degrees of freedom of the gates are cou-pled to those of the brownian motion of ions by means ofthe interaction potential VI(Yj , xi). A similar interactionwas also considered in reference [14] in the study of a Clchannel. By means of this term the ions see the gate as a

Eur. Phys. J. B (2018) 91: 10 Page 3 of 9

barrier potential. As we will see the fact that the modelis formulated by a single energy functional implies thatinteraction between ion and gate is mutual. In particular,gate dynamics could be affected by the collision with ions,which will be of capital importance in the results.

2.2 A and B pore models

We will treat two pore models, A and B, representativeof two generic channel families for the Na and K ions.Regarding the Na permeability, and following notation ofreferences [4,15], the H–H formulation considers two differ-ent modulated functions m(∆V ) (activating) and h(∆V )(deactivating). Accordingly A-type pore will have an acti-vation gate Y1 and an inactivation gate Y2. B-type porewill have only one activation gate Y3, corresponding tothe modulating function n as the K channel in the litera-ture [15]. Results for these two pores will be comparedto available experimental data on Na and K channels.This theoretical scenario is complemented with the tableof parameter values used to work in the appropriate bio-logical scale (Tab. 1). Graphical representation of the gateeffective potentials for both models are shown in Figure 1.

3 Results and discussion

3.1 Pore gating

A relevant experimental information on the channel gat-ing is the probability of being open as a function of themembrane potential, Po(∆V ) [16]. We will evaluate thisprobability for both A and B models by performing sim-ulations of ion fluxes and gating dynamics by using theLangevin equations (2) and (3) for ions and gates at fixedvalues of the membrane potential, as it is done experi-mentally. The gate parameters have been chosen to exhibitthe experimental gating properties: flux, gating dynamics,time scales, etc. (see Tabs. 1 and 2).

3.1.1 Pore A: Y1 and Y2 gates

We proceed with the study of the two gates of the Amodel.

Gate Y1. The two equations (2) and (3) for j = 1 intro-duced in the previous section are simulated. The secondgate Y2 of the channel is also simulated and let fluctuate inits open state, but it is forced to never close. For each valueof the membrane potential we record a very long run of astochastic trajectory, which exhibits roughly rectangularrandom pulses corresponding to rapid switchings betweenthe two states Y1 ∼ 0, 1. The time spent in the open state,t0, divided by the total time tt gives the open probabil-ity for this potential, Po(∆V ) = t0/tt. In Figure 2 – top(black dots) we see the results. The empty dots correspondto the case without ions (zero concentrations) and the fulldots to the presence of concentrations.

Gate Y2. This gate is simulated by the same procedure,but this time it is the Y1 gate which is forced to remainin its open state. Numerical simulation results are plottedin Figure 2 – top (red dots). Empty dots correspond to

Table 1. Physical parameter values used in the sim-ulations. Both ions have a positive charge q = +1 e.

γA particle friction 2µs meV/nm2

γB particle friction 8µs meV/nm2

KBT 25 meVL channel length 4 nmA channel section 4 nm2

cA0 (in), cA1 (out) 0.092, 0.5 McB0 (in), cB1 (out) 0.54, 0.075 MCM effective capacity 1.25 charges/mV

Fig. 1. Gate potentials for ions corresponding to both channelmodels A and B, evaluated for the steady values of closed gatesY1 = 0.029, Y2 = 0.022 and Y3 = 0.029.

the gate with no ions and the full dots with Na concentra-tions. The Po(∆V ) of this gate has the same qualitativeS-like plot than that of Y1 but with the steady statesinterchanged because effective charge has a different sign.

3.1.2 Pore B: Y3 gate

We proceed with the model B following the same proce-dures as in the previous case. This time Y3 is the only gatepresent in the pore. Numerical results are seen in Figure 2– bottom (black squares). Empty symbols correspond tothe simulations without ions and full symbols to simula-tions in the presence of ionic concentrations. We see thatY3 behaves as Y1 but with different parameters. As in theformer pore we observe important differences due to thepresence of ionic concentrations, and the direction of thiseffect is the same as in the Y1 case.

We then see in all gates a clear effect of the presence ofions to favor the open state. This result is a consequence ofthe physical consistency of the approach, in particular ofthe fact of using a single energy functional for the mutualinteraction between ions and gates. Gates exert forces onthe ions which implies that they will also have an effect onthe gates. As a result the S-like P0 plot moves to smallerpotentials for activation gates, and to larger potentials forinactivation gates.

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Table 2. Parameters of the Y1, Y2, Y3 pore gates usedin simulations. Units for γ are µs meV/nm2 and σ =0.283 nm.

γ V0 Vd Q φref a b xc(kBT) (kBT) (e) (mV) (nm)

Y1 1000 7 8 +12 −35 0.2 7 1.0Y2 4000 7 10 −8 −35 0.2 9 3.0Y3 4000 7 8 +10 −35 0.2 7 3.0

Fig. 2. Voltage gating characterization of A and B modelgates. Top: pore A, Y1 (black) and Y2 (red) gates. Bottom:pore B, Y3 gate. Empty symbols: simulations of only gates,i.e. without ions. Full symbols: simulations with ion concen-trations as in Table 1. Lines are the fits of expression (6) withthe parameters values of Tables 1–3.

Moreover, in the case of model B we observe that thiseffect mainly acts by making the closing slower. This canbe seen in Figure 3, where the average of the staying timesin each state are shown in the same cases as in Figure 2 –bottom. We see that the effect of ions on the closed gateis rather small, since there is only a marginal reductionof the stay time for the smaller polarizations. Howeverthere is an important increasing of the stay times for theopen state with concentration. In other words the channelcloses more slowly in the presence of ions. This effect, wellknown experimentally [17], is often attributed to a “foot-in-the-door” mechanism. Note that no specific mechanismhas been added to the model to provide such effect. Onthe contrary it has naturally emerged from the physicalapproach used here.

We return now to the open probability obtained inFigure 2. The function Po(∆V ) is known from well con-trolled experiments on both single and ensemble of chan-nels. The standard explanation for the shape of thisfunction [1,3,16] is that a voltage gating gate has twosteady states, open and close, which have different energiesUO(∆V ) and UC(∆V ) respectively. Then the character-istic times are weighted by the corresponding Kramersfactor for the crossing of a barrier tU ∼ exp−U/kBT .The relative temporal fraction of an open state, i.e. the

Fig. 3. Dynamic characterization of gating for the B model.Blue circles: mean time in the closed state. Black squares:mean time in the open state. Empty symbols: simulations ofonly gates, i.e. without ions. Full symbols: simulations with ionconcentrations as in Table 1.

Table 3. Gate physical parameters. Second and thirdcolumns are the parameter values used in the simula-tions, shown for comparison (see Tab. 2). The next twocolumns (o) are their effective values without ion effects,and the last two columns (c) are the effective values whenion concentrations are present. All these effective valueshave been obtained from simulation results.

Q φref Qoeff φo

eff Qceff φc

eff

(e) (mV) (e) (mV) (e) (mV)

Y1 +12 -35 +10.72 −35.01 +10.28 −37.86Y2 −8 −35 −6.88 −34.56 −6.94 −30.94Y3 +10 −35 +7.89 −35.08 +8.88 −37.86

probability Po is then,

Po(∆V ) =e−UO/kBT

e−UO/kBT + e−UC/kBT

=1

1 + e− ∆U

kBT

=1

2

(1 + tanh

∆U

2kBT

), (6)

where ∆U = UC − UO = Qeff(∆V − φeff).This expression fits very well with the experimental

data and accordingly it permits to fix the model inter-nal parameters Q and φref. Nevertheless comparing thevalues used in our simulations with those obtained by thefitting of the simulation results (Qeff and φeff in Tab. 3),we observe important differences. The origin of thesediscrepancies is due first to the Kramers mathematicalapproximation and second to the presence of concentra-tions. In our simulations these three gates have the sameφref = −35 mV, but different gating charges. The fit ofequation (6) gives different effective values for Qeff and φeff

depending on whether ions are present (c) or not (o). It ismanifest that the Kramers approximation mainly affectsthe effective charges and the concentrations change theeffective potential.

Eur. Phys. J. B (2018) 91: 10 Page 5 of 9

Fig. 4. Current peak for A pore (red) and mean current for Bpore (blue) versus the applied voltage ∆V . Lines correspondto the analytical expression (7).

3.2 Mean ionic fluxes

By using the effective parameter values of Table 3 one cancalculate the flux of our models from the GHK equation(A.6) for a pore but modulated by the expression for Po

in equation (6),

I(∆V ) = P0(∆V )∆V ρinγxL

×e(VNernst−∆V )/kBT − 1

e−∆V/kBT − 1CIJ , (7)

where CIJ = 0.1602 pAµs/n is a conversion factor fromparticle flux J to intensity I. In Figure 4 we see simula-tion results (symbols) from the whole model together withtheoretical predictions (lines) from equation (7). For theB pore (blue symbols) we plot the numerical steady flux,and for A pores (red symbols) we represent the currentpeak. These results are strongly equivalent to the exper-iments of reference [18] for Na (model A in our model)and K (model B) molecular channels. One can concludethat GHK equation together with our approach constitutean appropriate physical scenario to explain experimentalresults.

3.3 Dynamics of single pores

We progress further with the study of the behavior of asingle pore, for both A and B cases, under a depolarizingpotential step. Analogous experiments have been key forthe understanding of the internal structure and conforma-tions of the channel, and therefore it is worth to comparethe simulations of the model to the available experimentalresults.

We start with the numerical simulations of the dynam-ics of a single A pore under the perturbation of depolar-izing voltage steps from −90 mV to −10 mV. We expectour A model will mimic the experimental results for Nachannels [2,19]. Four square pulses are seen in Figure 5

Fig. 5. Pulses on the single A model. From top to bottom.A depolarizing step from −90 mV to −10 mV applied to thepotential membrane during 40 ms. Fluxes of four pulses underthis step, and the average of 200 events. Intensity signals arefiltered by an averaging window of 0.125 ms.

Fig. 6. Dynamical evolution of Y1 and Y2 variables (gates)during the second pulse of Figure 5.

during depolarization steps. After the depolarization thegate-1 opens randomly in a short time scale, then after alarger random time interval gate-2 is closed. Within thisinterval ions are able to cross the membrane and a largerintensity is observed. The mean average of 200 pulses isa spike-like pulse with fast growing and slow decay. Thisis the behavior observed in Na single channel experiments(see Fig. 2 in Ref. [19]).

In the literature these results are explained by assum-ing that the Na channel has three possible states: open,close and inactive [2]. In our channel A we have four pos-sible states: (Y1, Y2) ' (0, 1), (1, 1), (1, 0), (0, 0), shown asa, b, c, d in Figure 6. These states can be classified as: closebut ready (standby) a = (0, 1), open b = (1, 1), and thelast two, with Y2 ' 0, correspond to inactive refractorystates c = (1, 0) and d = (0, 0). In this figure we see howthe duration of the open state, b, is quite random andthat the last one, d, is very short in time. The temporal

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Fig. 7. Pulses on the single B channel. From top to bottom.A depolarizing step from −90 mV to −30 mV applied to thepotential membrane during 20 ms. Fluxes of four pulses underthis step, and the average over of 200 pulses. Intensity signalsare filtered by an averaging window of 0.125 ms.

evolution of these variables, going through these states inone depolarizing event is seen in this figure, which corre-sponds to the second pulse in Figure 5. The three closestates could be experimentally discriminated by a fineranalysis of the different intensities of the leak flux. Thisconstitutes an additional prediction of the model.

The same procedure is next applied to a single B porewith depolarizing voltage steps from −90 mV to −30 mV,as it is seen in Figure 7. This last voltage corresponds toa high probability for the open state. Numerical resultsshow that the gate opens quite randomly but it remainsopen almost all the time until the end of the perturbation.This behavior is similar to that observed in experiments ofK single molecular channels (see Fig. 4-A from Ref. [20]).

4 Concluding remarks

We have applied a semi-microscopic Langevin approach tostudy the gating dynamics of single ionic channels mod-eled as active pores. The approach is characterized bythe use of a few simple physical mechanisms, variables,and physical laws in a consistent way. This is enoughto explain the experimental information concerning gat-ing dynamics, ion fluxes, membrane potential and ionicconcentrations. Thus we do not need to consider furtherbiological complexities that would be necessary for morespecific observations, such as for example ionic selectivity.

It has been assumed here that ions are charged Brow-nian particles which follows Langevin equations follow-ing standard statistical physics. These pores have gatesexhibiting two steady states (open and close) whosedynamics are controlled by nonlinear elastic potentials

and Langevin equations. Ions and gates are treated asmechanical objects interchanging energy and momentumby mutual collisions, which is accomplished in the modelby using a single energy functional. The number of degreesof freedom used is thus maintained to a minimum. Thispermits to isolate the relevant mechanisms for the stud-ied phenomenology and to obtain results comparableto experiments without the need of additional detailedchannel structure. Moreover the physical nature of theapproach permits, in a straightforward way, its extensionfor a more quantitatively detailed study and for takinginto account additional mechanisms in other channels oralternative gating modelings.

We have presented two pore models representing Naand K molecular channels. Their gating dynamics, as rep-resented by the open probability of the pores and bythe steady and peak ionic currents as functions of mem-brane potential, exhibit the main characteristics observedin experiments on single channels [1,2].

Additionally, we have obtained the effect that concen-tration has on channel gating. Namely, the increasing ofion concentration enhances the probability of the openstate. Thus the open-probability curves move to smallervalues of the membrane potential for activation gates, andto higher values for inactivation ones. Very similar effects,associated to the so called “foot-in-the-door mechanism”,have extensively been studied experimentally, mainly inK channels [17] but also in other channels [21,22]. This isa most relevant result of our model, and it is particularlyinteresting the fact that it is a direct consequence of hav-ing a single energy functional for the mutual interactionbetween gates and ions, which constitutes a necessity ofphysical consistency. Therefore this effect should have ageneral validity for permeant ions, not depending on thespecific structure of the channel.

This approach opens interesting perspectives. Since allthe model parts are described by standard and well con-trolled physical laws one can address particular aspects forsingle channels or single gating events. Moreover model Aimplies the existence of three closed gate states for the Nachannel, which could be discriminated by measuring leakcharge intensities through the channel. This predictioncalls for experimental verification.

Channel parameters are obtained from experimentaldata and can be different for channels of the same fam-ily. These differences can enlighten a variety of internalchannel structures which can help to refine moleculardescriptions. In other situations different mechanical typesare possible. Also, we have shown the specific effect ofthe ionic cell concentrations on the gating process of eachgate. Our results show the interest in exploring furtherthis effect.

Further numerical refining could consist of includingion–ion interactions, which were omitted for simplicity,and due to the fact that it was not necessary to repro-duce the basic phenomenology. Also several additionalactivation gate variables could be implemented for eachchannel, to account for the tetrameric structure of fourvoltage sensing domains in the Na and K channels [2].These and other possible details, such as more complexpotential landscapes for both ions and gates [13], are

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straightforward in this framework and could be importantin order to use it for a more quantitative modeling.

Finally our approach can be extended to other possi-ble channel configurations, such as to other channels fromthe Na+ and K+ families, and also to model Ca2+ orCl− channels. Also, by using this approach, it would beinteresting to introduce ionic diffusion out of the mem-brane, in order to study the coupling among regions ofthe membrane [23] or with cell vesicles [24].

This work was supported by the Ministerio de Economia yCompetitividad (Spain) and FEDER (European Union), underprojects FIS2015-66503-C3-2P/3P and by the Generalitat deCatalunya Projects 2014SGR1093 and 2014SGR878.

Author contribution statement

J.M.S. designed research and performed the analytical cal-culations. L.R.-P. designed and performed the stochasticnumerical simulations. Both authors contributed equallyto the formulation of the approach, the discussion of theresults and to the writing.

Appendix A: Semi-microscopic approach

A.1 The electrostatic potential Vi(xi,∆V )

The theoretical framework for the ion dynamics is theknown Goldman–Hodgkin–Katz (GHK) equation [12], forthe flux through a pore, that we summarize here. Freeions inside the pore are described by point-like particlesof charge q moving in a one dimensional space at positionxi(t) under the effective electrostatic membrane potential

Vi(xi,∆V ) =q∆V

L(xi − L), 0 < xi < L, (A.1)

where ∆V is the potential difference between both sidesof the membrane, and L is the length of the pore.

A.2 The gate potential Vj(Yj,∆V )

The evolution of this variable is controlled by the nonlin-ear elastic potential,

V (Y,∆V ) = V0

[−a ln(Y (1− Y ))− b(Y − 0.5)2

]+Q(∆V − φref)Y. (A.2)

The first part has the form of a double well potential,and refers to the internal structure of the pore responsiblefor the gate bistability. The parameter values V0, a, b haveto be chosen to enter into the experimental scale. Notethat by choosing a � b the well minima are very closeto 0, 1, specifically at Y ' a

b , 1 −ab . A similar approach

was used in [21,25] for the gate in a Cl channel. Otherexpressions for the potential can be used, since the specificform is not really important, provided it has two minima,corresponding to open and close states, separated by anenergy barrier. Other forms for this double well potentialhave been explored, giving essentially similar results. Note

that the only relevant physical parameter of the bistablepotential is the height of the barrier.

The last term is the interaction with the membranepotential where Q is the charge of the gate sensor andφref is the reference potential that determines the ∆Vvalue at which both states are equally probable. Theselast two parameters are characteristic of a specific chan-nel and their values have to be obtained from experimentaldata (see for example experiments in Ref. [16]).

A.3 The ion–gate interaction potential VI(Y,xi)

The interaction between ions and gate is modeled by apotential energy, which represents a physical barrier forthe ions. This barrier has a specified position inside thechannel and a prescribed width. Its height is variable andcontrolled by the state of the gate represented by the Yvalue. The proposed potential is

VI(Y, xi) = Vdf(Y ) exp

(− (xi − xc)2

2σ2

). (A.3)

Here, xi is the ion position, xc is the the center of thegate potential inside the channel, and σ is its width. Notethat, in view of equations (2) and (3), this potential willproduce mutual forces on both ions and gate.

The height of the barrier is modulated by the functionf(Y ), which defines a correspondence between the stateof the gate (the Y value) and the height of the barrierVdf(Y ) seen by the ions. Thus the necessary conditionsfor the function f(Y ) are: f(0) = 1 for the close gate(maximum barrier height), f(1) = 0 for the open gate(minimum barrier height). Additionally, since Y is a fluc-tuating variable, we construct f(Y ) as having zero slopesat these values so the barrier height does not present largefluctuations while the gate remains in the same state.For the modulating function f(Y ) the envelope functionf(Y ) = (1 + cosπY )/2 is used. Other expressions for thisfunction have been tested, leading to very similar results.

A.4 Numerical methods

Our whole system is composed of three physical domainsor volumes: the channel, along which the ions move andwhere the gates are placed, and two reservoirs at bothchannel ends, corresponding to the regions inside andoutside the cell respectively.

The dynamics inside the channel are driven by Langevinequations (2) and (3). The numerical integration of theseequations is performed with a standard explicit (first orderEuler) algorithm. The employed time step has been ∆t =1.25 × 10−4 µs for the complete system (gates and ions)and ∆t = 10−2 µs when we had only gates (zero exter-nal ion concentrations). In particular ions move througha pore of length L, and they disappear from simulationwhenever the position xi escapes from the interval (0, L).

The reservoirs are implemented as boundary conditionsat both ends of the pore for the Langevin dynamics ofions. That means that the ionic concentration values atthe reservoirs determine the rate at which ions enter intothe pore.

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The numerical simulation of these equations allows torecord the evolution of the state of the gates and the num-ber of particles crossing the boundaries. In those caseswhere the membrane potential is not externally fixed themembrane potential ∆V is updated by using the numer-ical integration of the capacitor equation (Eq. (5)) in theform

∆V (t+ ∆t) = ∆V (t)− ∆Q1 + ∆Q2 + ∆Qext

2CM, (A.4)

where ∆Q1 and ∆Q2 are the total charge crossing each ofthe two boundaries in the time interval ∆t. The additionalterm ∆Qext accounts for external inputs to the membranecharge. The divisor “2” takes into account that chargeswhich cross both boundaries are being counted twice. Thefinal output are the values of the membrane potential∆V and of the ionic fluxes J = ∆Q/∆t, which will becompared with known experimental results.

A.5 Validation of the stochastic approach

A necessary test of the model is the dynamical relaxationof the open channel to its steady state in the presence offixed ion concentrations at the boundaries. This is donefor both models. The evolution of the membrane potentialto the Nernst value corresponding to the concentrationsat both sides of the membrane (without any parameterfitting) supports the consistence of the dynamical model,and also the correctness of the used algorithms.

From the theoretical point of view ions obey theoverdamped Langevin equation,

γxxi = −∂xiVi(xi) + ξi(t), (A.5)

where γx is the effective friction and ξi(t) is a ther-mal noise of zero mean and correlation 〈ξi(t) ξi(t′)〉 =2γxkBTδ(t− t′). One can consider here the Fokker–Plankequation for the density of ions inside the channel. Assum-ing steady state with constant flux, J(x, t) = J , and theboundary conditions for the one-dimensional ionic densi-ties at both sides, ρin and ρout, we get the well knownGHK equation,

J(∆V ) =q∆V ρin

γxL

eq(VNernst−∆V )/kBT − 1

e−q∆V/kBT − 1, (A.6)

where VNernst = (kBT/q) ln cin/cout is the Nernst poten-tial corresponding to these concentrations, and q is theion charge. Note that the one-dimensional densities ρin/out

are related to the bulk concentrations cin/out as ρin/out =A cin/out, where A is the effective section of the channel.This equation is a relevant analytical reference when thechannel is in the open state.

In the simulations we initially populate the pore lettingions evolve with a fixed ∆V = 0 (this would correspondin experiments to electrically connecting both membranesides). At t = 125µs the voltage is then left free toevolve. From the stochastic evolution of the charges werecord the balance ∆Q crossing each of the channel

Fig. A.1. Time evolutions (colored curves) of the membranepotential induced by the ions of models A (upper curves)and B (lower curves) obtained from stochastic simulations.The potential membrane evolves to the corresponding Nernstpotentials VA ' 42.41 mV and VB ' −49.43 mV. Full blacklines correspond to the numerical integration of equation (A.7)and broken lines indicate the asymptotic Nernst values (seetext).

boundaries during the time step ∆t. That includes par-ticles hopping out of the system and particles enteringinto it through that boundary. Then from equation (5)the change in the membrane potential is evaluated andrecorded. Two realizations of each channel model areshown in Figure A.1.

This simulation result is complemented with the theo-retical calculation of the membrane potential from capac-itor equation (5) and the GHK equation (A.6), whichresults in the differential equation

d∆V

dt= −J(∆V )

Ceff. (A.7)

The numerical integration of this equation provides aprediction for the deterministic evolution of ∆V (t). TheNernst potential is the final steady value. This resultsshould be compared to the stochastic evolution obtainedfrom the numerical simulation of the Langevin equations.This can be seen in Figure A.1, where two stochastic evo-lutions of ∆V (t) are shown for each model (A and B). Itis worth to remark that they are single trajectories with-out any statistical average. These evolutions behave asexpected for both channels. In view of equation (A.7) thedifference in the times scales for both ions correspond tothe difference in the friction coefficients γx and in the ionconcentrations present in each channel.

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